Natural Frequencies Of Free Vibration Research Articles

Formulas for Calculating Deformations and Natural Frequency of Free Vibrations of a Hexagonal Tower

Statement of the problem. A scheme of a six-sided prismatic statically determinate spatial girder is proposed. The task is set by induction to derive formulas for the dependence of the deflection of the structure and the lower limit of the main frequency of natural vibrations on the number of panels along the height of the prism. Materials and methods. The forces in the rods along with the reactions of the supports are found in an analytical form by means of the method of cutting nodes in the Maple symbolic mathematics system. One of the nodes at the base of the truss has a spherical support, one has a cylindrical support, the remaining four supports are racks. The top deflection is determined by the Maxwell-Mohr formula. From the analysis of the sequences of coefficients in the formulas for individual structures with a different number of panels, their common members are determined, which are included in the desired calculation formula. The Dunkerley method is used to estimate the first frequency of free oscillations. Results. For various types of loads, formulas for the dependence of girderf deflections on the number of panels are obtained. The coefficients in the solution are polynomial in the number of panels. The derived analytical dependence of the first frequency on the number of panels in comparison with the numerical solution has a small error, which decreases with increasing number of panels. Conclusions. A design of an axisymmetric statically determinate tower-type girder has been developed, which allows analytical solutions to the problem of deflection and the problem of the first natural frequency for an arbitrary number of panels. The resulting formulas can be used to assess the accuracy of numerical solutions and for preliminary calculations of models of structures of this type.

Residual stress dynamic detection and vibration characteristics of wire saw busbar

Diamond wire saw cutting is the primary method for processing single crystal silicon carbide (SiC) and silicon (Si) in the photovoltaic and semiconductor industries, the diamond wire saw is formed by busbar electroplate diamond particles, and the diamond wire saw busbar is drawn from steel wires, and the residual stress generated during the drawing process is a critical factor to determine whether the busbar is easy to break. This paper proposes a method to assess residual stress by measuring the free vibration frequency of the busbar, which detects residual stress of busbars with extremely fine and ultra-long (lengths of up to 250 km and diameters as small as 0.04 mm) for real-time, accurate, non-destructive way. First, the residual stress distribution of the drawn busbar is obtained by Abaqus simulation, according to the simulation results, this paper analyzes the mechanical mechanism underlying different types of free bending in the busbar, and develops the relationship between the free bending circle diameter and the residual stresses; Then, the residual stresses are equated with the axial tensile stresses of the busbar by numerical methods, the equivalent mathematical model of the stresses is developed, and the correctness of the model is verified by experiment. Finally, the dynamic model of the moving busbar with residual stresses is established, the effects of different tensions and velocity on the free vibration characteristics of the busbar under residual stresses are analyzed, and the correctness of the detection method is verified by numerical calculations and experimental comparisons. The results show that the residual stress of busbar drawing can be equivalent to tensile force numerically, and the larger the residual stress, the smaller the circle diameter of the busbar, and the larger the first-order natural frequency of free vibration.

Axisymmetric thermal vibration analysis of graded porous circular plates based on physical neutral surface and Reddy’s third-order shear theory

In this article, the free vibration of functionally graded porous (FGP) circular plate with various boundary conditions are investigated by means of the differential quadrature method (DQM) based on Reddy’s theory. Performance of the materials varies continuously in the whole thickness and two cosine forms of nonuniform porosity distribution along its thickness are considered. Hamilton’s principle is adopted to derive the governing equation of the system, which effectively considers the effects of thermal stress. It is worth noting that due to the introduction of physics neutral surface, reducing stretching-bending coupling effect, so the complex governing equations have been appropriately simplified. Then, using the DQ method, the natural frequencies of free vibration of FGP circular plates subjected to thermal environments with various boundary conditions were obtained. Convergence and comparative research are performed to prove the convergence, reliability, and accuracy of the DQ method. The effects of various factors such as porosity coefficient, porosity distribution pattern, temperature rise, thickness-diameter ratio, and boundary conditions on the natural frequencies are discussed in detail.

Estimation of Natural Frequencies of Cantilever Beam with Open Inclined Edge Crack

This paper presents an estimation of natural frequencies for free vibration of a cantilever beam in the presence of an inclined edge crack using linear elastic fracture mechanics. The Strain Energy Release Rate (SERR) approach is employed to compute the compliance coefficients for the edge crack inclined to the longitudinal axis at an arbitrary angle. A full compliance matrix under the general loading is presented and corresponding definitions of the compliance coefficients are given. Calculations of natural frequencies are carried out for both Timoshenko and Euler-Bernoulli beams under the influence of load P1, P3, and P5. Contribution to strain energy coming from various modes of fracture is accounted for by using stress intensity factors obtained numerically using ABAQUS 6.14 software. The effect of the crack on transverse and longitudinal natural frequencies is studied for different crack locations, depths, and inclination angles. Experiments are performed on short and long beams for a set of parameters for comparison purposes. It is found that the natural frequencies estimated by the proposed method compare well with the frequencies obtained experimentally for both beams.

Free Vibration Analysis of Laminated Composite Plates Based on the Reconstructed Edge-Based Smoothing DSG Method

In order to avoid transverse shear locking and to improve the accuracy, the RES-DSG3 method is proposed by incorporating a non-isoparametric DSG method with a novel smoothing technique based on global coordinates, and all the integration of smoothed matrices can be calculated along the boundary segments of smoothing cells without coordinate mapping. Based on the first-order shear theory, the RES-DSG3 method is used to analysis free vibration natural frequencies of composite laminates from different material parameters, edge thickness ratio and boundary conditions. The free vibration of composite laminates is analyzed numerically. The calculation of the example verifies the feasibility and effectiveness of the method.

Compensatory bellows oscillations as a corrugated shell with liquid inside

The paper deals with a relevant problem of shipbuilding, i.e. calculation of free and forced vibrations of pipeline compensatory bellows. These devices are used to reduce the vibration load caused by ship power machines. When analyzing the vibrations of the compensatory bellows, it is necessary to take into account the liquid contained in the bellows. In this work, the design model of the bellows is represented by a corrugated elastic shell as a material surface with five degrees of freedom. A variant of the classical theory of shells, built on the basis of Lagrangian mechanics, is used. The influence of the liquid is taken into account by two models. First, the liquid is considered to be ideal and incompressible and is considered through the attached mass to the shell. The shell is replaced by a cylindrical surface with a radius in the middle line of the corrugation. To account for the influence of the frequency of bellows oscillations on the attached inertia of the liquid in the calculation we also used the acoustic approximation; and derived a formula for a generalized attached mass of the ideal compressible liquid. The equations of the bellows oscillations under the periodic loading are obtained. The problem has been solved by the finite difference method. The values of natural frequencies of free vibrations are obtained for the compensatory bellows from the corrosion-resistant heat-resistant steel. It is shown that by taking account of the liquid, we significantly change the natural frequencies of the bellows. With high-frequency vibrations it is necessary to take into account the compressibility of the liquid. The problem of the forced vibrations of the bellows caused by a displacement of its end face by the harmonic law is solved. The internal forces and moments are determined, as well as occurring stresses by Mises criterion in the bellows. We found the critical value of the end face displacement at a frequency of 50 Hz, at which the bellows goes into a plastic state.

Flexoelectric effects on the natural frequencies for free vibration of piezoelectric nanoplates

Flexoelectricity is an electromechanical coupling phenomenon between polarization and strain gradient. Based on the Kirchhoff thin plate theory, the electromechanical coupling responses of nanoplates with the piezoelectric and flexoelectric effects are studied in this paper. Free vibration of a piezoelectric nanoplate with consideration of flexoelectricity is analyzed with emphasis on the influence of the dynamic flexoelectric effect on the natural frequencies. By means of Hamilton’s variational principle, the governing equation of rectangular plates together with associated boundary conditions is derived. The natural frequencies are evaluated for a nanoplate simply supported at two opposite edges, and exact frequency equations are obtained for the other two opposite edges being simply supported, clamped–clamped, clamped–free, simply supported–free, or clamped–simply supported. The influence of dynamic flexoelectricity on the natural frequencies is elucidated. The results show that the dynamic flexoelectric effect is also size-dependent; the smaller the plate thickness is, the more obvious the dynamic flexoelectric effect is. The results also show that the dynamic flexoelectric effect is more pronounced when the order of vibration modes is higher and nanoplate’s side ratio is larger. The positive and negative choice of static and dynamic flexoelectric coefficients have completely different effects on the natural frequencies. The influence of the dynamic flexoelectric effect on the natural frequencies is closely related to the side constraint and geometry of the plate. The piezoelectric effect does not alter the natural frequencies for free vibration of a homogeneous nanoplate.

Differential Quadrature Method for Vibration Analysis of Prestressed Beams

This paper investigates natural frequencies of free transverse vibrations of prestressed beams, and the governing equations and natural frequencies of the free vibration in related literatures are discussed and corrected. The differential quadrature methods (DQ) are applied directly to the corrected governing equations to get the the values of natural frequency numerically. Under the simple supported boundary conditions, the natural frequencies of model beam are numerically studied, and the physical parameters of the beam are analyzed respectively. The numerical results show that the natural frequency values increase with the growth of concrete strength and eccentricity of prestressed steels. But with the increase of the span length of beam and values of original prestressing force, the natural frequency values decrease.

Vibration Modelling and Control Experiments for a Thin-Walled Cylindrical Rotor with Piezo Patch Actuation and Sensing

This paper describes a dynamic model formulation and control experiments concerning the vibration behaviour of a thin-walled cylindrical rotor with internal piezoelectric patch transducers. Model development, validation and controller design procedures were undertaken for an experimental rotordynamic system comprising a tubular steel rotor (length 0.8 m, diameter 0.166 m and wall-thickness 3.06 mm) supported by two radial active magnetic bearings. Analytical solutions for mode shapes and natural frequencies for free vibration were first derived using a shell theory model, and these used to construct a speed-dependent parametric model for the rotor structure, including piezo patch actuators and sensors. The results confirm that the developed shell theory model can accurately capture the rotating frame dynamics and accounts correctly for frequency splitting from Coriolis effects. The model is also shown to be suitable for active controller design and optimization. Model-based H2 feedback control using the rotor-mounted actuators and sensors is shown to achieve vibration suppression of targeted flexural modes, both with and without rotation.

Numerical approach of free and forced elastic vibrations using high-regularity Hermitian partition of unities

In this paper, shape functions with regularity up to $$C^{2}$$ were developed for the four-node quadrilateral finite element considering the partition of unity property. This high-regularity approximation space was applied to approach the natural frequencies of free vibration of some in-plane elastic problems as well as the elastic wave response of forced vibration caused by the application of impulsive loading. The obtained results show that it was possible to numerically approximate a greater number of accurate natural frequencies with the devised procedure when a comparison is established with the $$C^{0}$$ Lagrangian and serendipity elements of 4, 8 (serendipity), 16, and 25 nodes. Also, the numerical predictions for the elastic wave propagation problem using the derived approximation space presented small oscillations improving the representation of the displacement field.

Generalized integral transform solution for free vibration of orthotropic rectangular plates with free edges

Free vibration of orthotropic rectangular thin plates of constant thickness with two opposite edges clamped and one or two edges free is analyzed by generalized integral transform technique. Numerically stable eigenfunctions in exponential function forms of Euler–Bernoulli beams with appropriate boundary conditions are adopted for each direction of the plate. The governing fourth-order partial differential equation for the mode function of free vibration is transformed into a system of linear equations, by integral transform in both directions of the rectangular plate. The boundary conditions at free edges are satisfied exactly by considering the terms generated in the transformed equations by integration by parts, which are absent in the equations by traditional Rayleigh–Ritz method. The natural frequencies of free vibration of orthotropic rectangular thin plates obtained by the proposed integral transform solution are compared with available results in the literature and numerical solutions by finite element analysis, showing excellent agreement and high convergence rate.

Nonuniform Isospectrals of Uniform Timoshenko Beams

Spectrally equivalent systems are those that have the same free vibration natural frequencies for a given boundary condition. In this paper, we establish isospectrality between certain classes of nonuniform Timoshenko beams with a given uniform Timoshenko beam. We apply a transformation to convert the nondimensional coupled nonuniform Timoshenko beam equations from the frame of reference to a hypothetical frame of reference. The transformed equations are then combined by eliminating one of the variables. Specific material and geometric properties are chosen, and a few auxiliary variables are introduced to convert the transformed equation into the required form. If the coefficients of the transformed equation match with the required uniform equation, then the nonuniform beam is said to be isospectral to the uniform beam. The boundary configurations also change during this transformation. We present the constraints under which they are preserved. Frequency equivalence of the beams is confirmed by the finite element method. For the considered cases, examples of beams having a rectangular cross-section are also presented.

Bending and free vibration of a circular magnetoelectroelastic plate with surface effects

Surface effects play a significant role in affecting the mechanical behavior of nanosized NEMS such as sensor, actuator, transducer. This paper studies the bending and free vibration of a magnetoelectroelastic plate with surface effects. By incorporating surface effects into the Kirchhoff theory of thin plates, the governing differential equation for bending and vibration of a magnetoelectroelastic plate is derived. Simply supported and clamped circular plates are analyzed for applied mechanical loading, electric voltage, and magnetic potential difference. An analytical method is presented to obtain closed form solutions for static deflection of bending and natural frequencies of free vibration. The well-known results of circular thin plates are recovered if ignoring the surface effects along with coupling coefficients. A comparison of the deflections and the natural frequencies with their counterparts in the absence of surface effects shows that surface residual tension plays a crucial role, which decreases the deflection and increases the natural frequencies. The influence of surface magnetoelectroelasticity is discussed and the small scale effect can be captured. The obtained results are helpful for design and application of NEMS.

Vibration characteristics of cylindrical shells filled with fluid based on first-order shell theory

In this article, an analytical method to study the natural frequencies of free vibrations for a thick cylindrical shell filled with fluid is proposed. Mindlin’s first-order shell theory is extended to derive the equations of motion and corresponding boundary conditions by Hamilton’s principle. Linearized potential flow theory is used to derive the hydrodynamic force. Moreover, the internal fluid pressure acting on the shell wall is obtained by the assumption of a non-penetration condition. The dispersion equations are obtained under the assumption of harmonic motion. The derived shell theory is used to calculate the natural frequencies of cylindrical shells with various thicknesses and lengths, and the results are compared with Flugge’s shell theory and finite-element method (FEM). As a result, the proposed shell theory shows improved accuracy and good agreement with published experimental results.

A new approach for free vibration analysis of nonuniform tall building structures with axial force effects

SummaryDynamic analysis of beam‐like structures is significantly important in modeling actual cases such as tall buildings and several other related applications as well. This article studies free vibration analysis of tall buildings with nonuniform cross‐section structures. A novel and simple approach is presented to solve natural frequencies of free vibration of cantilevered tall structures with variable flexural rigidity and mass densities. These systems could be replaced by a cantilever Timoshenko beam with varying cross‐sections. The governing partial differential equation for vibration of a nonuniform Timoshenko beam under variable axial loads is transformed with varying coefficients to its weak form of integral equations. Natural frequencies can be determined by requiring the resulting integral equation, which has a nontrivial solution. The presented method in this study has fast convergence. Including high accuracy for the obtained numerical results as well. Numerical examples including framed tube as well as tube‐in‐tube structures are carried out in the study and compared with available results in the literature, and also with the results obtained from finite element analysis in order to show the accuracy of the proposed method in the study. Obtained results indicate that the presented method in this study is powerful enough for the free vibration analysis of tall buildings.

Free vibration analysis of nano-plate in viscous fluid medium using nonlocal elasticity

In the present study, the dynamics of the interaction between a Kirchhoff nanoplate and the surrounding fluid is investigated. Using non-local elasticity theory, the influence of small-scale parameter is considered in the governing equation of motion. The Navier-Stokes equations are utilized to model the fluid-solid interaction. The vibration behavior of nanoplate submerged in different viscous fluids with various aspect ratios are simulated in order to analyze the effects of fluid viscosity and density on the free vibration natural frequencies of the nanoplate. The results reveal that, for smaller size parameter, the fluid existence has a remarkable decreasing effect on the nanoplate natural frequencies. Moreover, it has been shown that the fundamental frequency ratio between two cases, first, when the nanoplate is assumed to be under viscous fluid interaction, and second, while the structure vibrates in vacuum environment, remains almost constant with the changes in small-scale parameter.

On the Natural Frequency of Free Vibrations of the Gas Envelope Surrounding a Solid Particle or a Droplet

The behavior of two-phase bubbles containing a solid or a liquid particle in their interior has been studied. A simple formula for the frequency of free radial vibrations of such a bubble has been obtained.

Effects of axial movements of the ends and aspect ratio of laminated composite beams on their non-dimensional natural frequencies

This study developed to solve the problem of prediction of the natural frequencies of free vibration for laminated beams. The study presented the natural frequencies of composite beams with four layered and different boundary conditions. In each boundary condition, two cases are assumed: movable ends and immovable ends. Numerical results are obtained for the same material to demonstrate the effects of the aspect ratio, fiber orientation, and the beam end-movements on the non-dimensional natural frequencies of beams. Two aspect ratios are given in the numerical results, one is for relatively short-thick beams, while the other is for slender beams. It was found that the results of the non-dimensional frequencies obtained from the short-thick beams are generally much less than those obtained from the other slender beams for same fiber orientation and generally, the frequencies of longitudinal vibration increase as the aspect ratio increased. It was also found the values of the non-dimensional frequencies of the transverse modes are not affected by the longitudinal movements of the ends since these modes are generated by lateral movements only. However, the values of the natural frequencies of longitudinal modes are found to be the same for all beams with movable ends since they are generated by longitudinal movements only.

Non-rotating beams isospectral to rotating Rayleigh beams

In this paper, we establish spectral equivalence between certain non-uniform rotating Rayleigh beams and a given uniform Rayleigh beam, and also between non-uniform non-rotating Rayleigh beams and a given uniform rotating Rayleigh beam. Spectrally equivalent systems are those that have the same spectra, i.e., the same free vibration natural frequencies for a given boundary condition. We use a transformation to convert the non-dimensional Rayleigh beam equations from the (x, W) frame of reference to a hypothetical (z, U) frame of reference. If the material and geometric properties of the beam are specific chosen functions of the two introduced auxiliary variables, then the transformation will be achieved. If the coefficients of the transformed equation match with the required equation, then the equation, on which the transformation is applied, is said to be isospectral to the required one. The mode shapes for these particular cases also match due to the nature of assumption on one of the introduced auxiliary variables. Frequency and mode shape equivalence of the beams is confirmed by the finite element method (FEM). For the considered cases, examples of beams having a rectangular cross-section are presented to show the application of our analysis.

An analytical model for determining the effect of damping on 3D natural frequency of reinforced walls

Most of the proposed methods for obtaining the free vibration natural frequency of the retaining wall assume the behavior of the wall in the two-dimensional domain, and they are not able to express the three-dimensional behavior of these structures in a satisfying manner. Moreover, the effect of damping has not yet been considered in analytical study. In this paper, the plate theory and Galerkin method is employed to analyze the free vibration of a wall-reinforcement-soil system in the three dimensional domain and survey the effect of damping on the natural frequency of a retaining wall. So, the retaining wall is modeled as a clamped-free plate and the stiffness of the soil and reinforcements existing behind the wall are modeled as a set of springs and the damping effect of the system is modeled by a set of dampers located at the back of the Retaining wall and attached to it. The results of the proposed model are compared with both the results of the other researchers and the ones from finite element method (FEM). They are also compared with the results of a full-scale experiment and it shows a good agreement. The proposed method which uses the plate theory, also yields the free vibration natural frequencies of the wall’s extensional modes with an acceptable accuracy. Finally, the effect of tensile and compressive behaviors of the soil and the reinforcements on the fundamental frequency of a reinforced retaining wall is studied.