Free Vibration Frequencies Research Articles

A study of the length parameter and the longitudinal compressive forces in the determination of frequencies of free vibrations of thin-walled underground pipelines of large diameter

Earlier work on the study of frequency characteristics of thin-walled underground pipeline to obtain an equation for finding the natural frequencies of the straight sections of the pipeline, taking into account the parameter of the longitudinal force, the magnitude of internal pressure, coefficient of elastic resistance of the soil, the option of thin tubing and added mass of the soil. In this article, using the obtained equation, we study the influence of the length of the section of the pipeline laid in the soil with different physical-mechanical characteristics, and the effect of the parameter of the longitudinal compressive forces at the frequency of free oscillations of thin-walled straight pipeline under the action of various internal working pressure for pipes of different diameters with different wall thicknesses. On the basis of the design data defined in the derived formulae, is determined by the criterion of application of shell theory or the core theory for finding the natural frequencies of thin-walled underground pipelines of large diameter, depending on the length of the element. Simultaneously, the obtained expression allows to determine the critical force at which buckling occurs in the core of the theory («beam buckling»), as well as the formula to determine the critical force, from which buckling occurs by shell theory (flattening of the cross section). Based on these data, it is concluded that flattening the cross section of the pipeline will occur when the force is at times less than required for the formation of «arch release», and consequently to ensure the reliability of underground thin-walled large diameter pipeline should be the first thing to check for resistance on the shell theory.

Nonlinear component of free vibration frequency of a partially tensioned system with consideration of rotational stiffness of support

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

Investigation of Free Vibrations of Three-Layered Circular Shell Supported by Annular Ribs of Rigidity

The construction of a mathematical model and the development of an algorithm of free vibrations investigation in the three-layered circular shell with a light-weight aggregate supported by annular rigidity ribs are considered in paper. The hypotheses of Kirchoff-Lyav are accepted for external bearing layers of shell and for aggregate there is accepted the linear law of tangential displacements change by thickness. The boundary conditions of a shell region closed between the ribs are established. Using the boundary transition, conditions along the lines of the ribs, taking into account and without deformations of displacement in the ribs, but without taking into account the torsional rigidity in the ribs are determined. The equation of motion of supported three-layered shell is obtained. The frequencies of free vibrations were investigated and values of parameter of the first frequency of free vibrations for a shell, supported by one and three rigidity ribs, were calculated. There are given values depending on the physical and mechanical properties of materials and geometric dimensions of the shell, the curvature parameter, and the rigidity parameter of an aggregate.

A Stochastic Galerkin Cell-based Smoothed Finite Element Method (SGCS–FEM)

In this paper, the cell-based smoothed finite element method is extended to solve stochastic partial differential equations with uncertain input parameters. The spatial field of Young’s Modulus and the corresponding stochastic results are represented by Karhunen-Loéve expansion and polynomial chaos expansion, respectively. Young’s Modulus of structure is considered to be random for stochastic static as well as free vibration problems. Mathematical expressions and the solution procedure are articulated in detail to evaluate the statistical characteristics of responses in terms of the static displacements and the free vibration frequencies. The feasibility and the effectiveness of the proposed SGCS–FEM method in terms of accuracy and lower demand on the mesh size in the solution domain over that of conventional FEM for stochastic problems are demonstrated by carefully chosen numerical examples. From the numerical study, it is inferred that the proposed framework yields accurate results.

Gas turbine impellers free vibration study using the fem analysis

Due to the rather hard working process gas turbine engines rotor impellers are always heavily influenced by the unfavourable factors of the gas flow. So the close attention should be paid to the vibration processes in the rotor. More worse is the fact that the impeller consists of blades, which have a curvilinear geometric form. So they can’t be correctly described by the theory of plates or even by the plate finite elements. That’s why the problem can be solved only by the usage of space curvilinear finite elements on the base of which the mathematical model is built. The whole impeller can be considered as a cyclically symmetric system. Thus it can be divided into several sections, one of which would be studied. After the nonlinear FEM approximation of the aforementioned impeller’s section we receive a canonic matrix equation of the solid body vibration state. As in this study only the problem of the impeller free vibration is taken into consideration, than the matrix of damping and the vector of external forces are equal to zero. After making all necessary preparations, we find out the section free vibration modes and frequencies. All calculations were held for two types of boundary conditions that are chosen according to the impellers assembling schemes. Calculated results have been compared with the experimental data. As the divergence between them is less than 15%, than the developed mathematical model is adequate. The developed mathematical model and obtained results could be used for the gas turbine rotors forced vibration and stress-strain state study.

Some closed-form solutions for static, buckling, free and forced vibration of functionally graded (FG) nanobeams using nonlocal strain gradient theory

In this paper, static bending, buckling, free and forced vibration of functionally graded (FG) nanobeams are studied within the framework of the recently proposed nonlocal strain gradient theory and the Euler-Bernoulli beam theory. The material properties of nanobeam are presumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fraction of the constituents. The governing equation and the related boundary conditions are derived via the principle of the calculus of variation. In order to eliminate the axial displacement in the formulation, the concept of the neutral surface is adopted. Some analytical solutions are obtained for the static displacement, critical buckling load, free vibration frequencies and the dynamic displacement for the case of the simply-supported end condition. In the dynamic analysis, three different loading cases, a moving load with constant velocity, point and distributed harmonic loads, are considered, and Duhamel’s integration is utilized for obtaining the corresponding dynamic deflections. Several numerical examples are presented in figures and tables in order to examine the effects of the strain gradient and the nonlocal parameters, the gradient index, the excitation frequency and the moving load velocity on the mechanical behavior of FG nanobeam.

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.

Longitudinal and Torsional Vibration Characteristics of Boron Nitride Nanotubes

Boron nitride nanotubes (BNNTs) are new cousins of carbon nanotubes (CNTs), which alternately substitute boron and nitrogen atoms for carbon atoms entirely. Its excellent physical and chemical properties, such as excellent thermal stability, chemical inertness and strong acid corrosion resistance, make BNNTs products be favorable to machine, thus scientific activities focused on them have profound practical significance and great application prospect. However, the vibration characteristics of BNNTs is still the least understood aspects. Longitudinal and torsional free vibration of single-walled boron nitride nanotubes are investigated through a coupling atomistic-continuum model with Euler beam model and atomistic simulations. Euler beam model with the involved atomistic-continuum multiscale model is capable of capturing the size effect since physical meanings of the integration for involved higher order gradient tensors are corresponding to size effect. The analytical solutions obtained by the coupling method are compared with atomistic simulation and are found to be in a good agreement. Computational results demonstrate that fundamental frequencies are in an order of 1 THz for both cantilevered and clamped boron nitride nanotubes, independent of tubular chirality. The effect of tubular radius on fundamental frequencies of both longitudinal and torsional free vibrations are large when it is small, especially the latter. The error between coupling method and atomistic simulation generally decreases as the length/diameter ratio increases. For a given length/diameter ratio, tubular radius have a distinctly decreasing effect on the error no matter what the kind of constraint is. However, a small difference observed in the study of torsional free vibration is that the clamped constraint gives rise to a slight increasing effect on the error as the tubular radius increases, but limited to a small degree. The vibrational modes confirm that the difference is originated from the involved breathing vibration in both torsional and longitudinal free vibration.

Transverse Vibration of Sandwich Beams with Thermal-Induced Nonuniform Sectional Properties

AbstractThis paper presents an analytical study of the transverse vibration of sandwich beams considering the thermal-induced nonuniform cross-sectional properties. Free vibration frequency formula...

Dynamic instability of castellated beams subjected to transverse periodic loading

In this study, an analytical solution is developed for the investigation of free vibration, static buckling and dynamic instability of castellated beams subjected to transverse periodic loading. Bolotin’s method is used to perform the dynamic instability analysis. By assuming the instability modes, the mass, stiffness, and geometric stiffness matrices are derived using the kinetic energy, strain energy and potential of applied loads. Analytical equations for determining the free vibration frequency, critical buckling moment, and excitation frequency of castellated beams are derived. In addition, the influences of the flange width of the castellated beam and the static part of the applied load on the variation of dynamic instability zones are discussed.

Nonlocal nonlinear model of Bernoulli–Euler nanobeam with small initial curvature and its application to single-walled carbon nanotubes

Although the small-scale effect and initial curvature have significant influence on the nonlinear mechanical property of the nanobeam, there are few researches to consider both factors. On the basis of the nonlocal differential constitutive relation, this paper, for the first time, presents a nonlinear partial differential–integral equation model of Bernoulli–Euler nanobeam with a small initial curvature. Then the model is applied to research the static bending and the frequency of free vibration for the single-walled carbon nanotubes (SWCNTs). This study indicates that the static deformations are relate to the load direction due to the initial curvature. In addition, the initial curvature complicates the relation between the free vibration frequencies and vibration amplitudes.

Pump or coast: the role of resonance and passive energy recapture in medusan swimming performance

Diverse organisms that swim and fly in the inertial regime use the flapping or pumping of flexible appendages and cavities to propel themselves through a fluid. It has long been postulated that the speed and efficiency of locomotion are optimized by oscillating these appendages at their frequency of free vibration. In jellyfish swimming, a significant contribution to locomotory efficiency has been attributed to the effects passive energy recapture, whereby the bell is passively propelled through the fluid through its interaction with stopping vortex rings formed during each expansion of the bell. In this paper, we investigate the interplay between resonance and passive energy recapture using a three-dimensional implementation of the immersed boundary method to solve the fluid–structure interaction of an elastic oblate jellyfish bell propelling itself through a viscous fluid. The motion is generated through a fixed duration application of active tension to the bell margin, which mimics the action of the coronal swimming muscles. The pulsing frequency is then varied by altering the length of time between the application of applied tension. We find that the swimming speed is maximized when the bell is driven at its resonant frequency. However, the cost of transport is maximized by driving the bell at lower frequencies whereby the jellyfish passively coasts between active contractions through its interaction with the stopping vortex ring. Furthermore, the thrust generated by passive energy recapture was found to be dependent on the elastic properties of the jellyfish bell.

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.

Geometrically nonlinear analysis of Timoshenko piezoelectric nanobeams with flexoelectricity effect based on Eringen's differential model

• Extended theory of piezoelectricity is used to investigate flexoelectricity effect on vibrational behavior of nanobeam. • The von-Kármán strain–displacement relationship, electric Gibbs free energy and Hamilton's principle are employed. • Multiple scales method is used to obtain a closed form solution of derived nonlinear governing equations of motion. • The effects of changes in parameters on free vibration and postbuckling behavior of nanobeam are discussed comprehensively. • Theoretical range of flexoelectricity constant is estimated through buckling analysis . This study analyzes the nonlinear free vibration and post-buckling of nanobeams with flexoelectric effect based on Eringen's differential model. The nanobeam is modeled based on Timoshenko beam's theory. The von-Kármán strain–displacement relation together with the electrical Gibbs free energy and Hamilton's principle are employed to derive equations of motion. The nonlinear free vibration frequencies are obtained for pinned–pinned (P–P) and clamped–clamped (C–C) boundary conditions. Multiple scales method is employed to obtain the closed-form solution for the nonlinear governing equations. By employing this methodology, the natural frequencies of nanobeams are obtained and their post-buckling behavior is examined. The influence of nonlocal parameter, amplitude ratio, and input voltage on the top surface and flexoelectricity constant on nonlinear free vibration and post-buckling characteristics of nanobeam is investigated. In this paper, it is concluded that the flexoelectricity has a significant effect on free vibration of the beams in nano-scale and its effect has to be considered in designing nano-electro-mechanical systems (NEMS) such as nano- generators and nano-sensors.

Monitoring wear of gear wheel of helicopter transmission using the FAM-C and FDM-A methods

This article is devoted to observation results of gear wheel wear of two different types of helicopter transmission: reduction gear of Mi-2 helicopter engine as well as accessory gearbox (AGB) of Mi-24 helicopter using proprietary FAM-C and FDM-A methods. The substantial sensitivity and complementarity of the observed phenomena was established – it is possible to simultaneously observe a given pair of a gear wheel and also other mechanical points, such as, e.g. the worn out bearing of the main shaft of helicopter rotor. The authors based their deliberations on the results of laboratory tests and complex tests on helicopters during their operation. The article is addressed at subsequent destruction stages of a gear wheel in helicopter transmission. The authors paid high attention to the observed wear due to the so-called structural resonance observed in some helicopters, where the worn out bearing of helicopter rotor results in the destruction of the pair of gear wheels. The gear wheel equalizes its frequency of free vibrations with the damaged rolling bearing in the line of rotation axis. Due to this, occurs the synchronous acceleration and deceleration of rotational speed of the wheel under consideration, which leads to material fatigue wear at the root of two teeth. Such wear may contribute to gear teeth breakage and helicopter crash.

ПРОГИБЫ И ЧАСТОТЫ СОБСТВЕННЫХ КОЛЕБАНИЙ СОСТАВНЫХ МНОГОСЛОЙНЫХ КВАДРАТНЫХ ИЗОТРОПНЫХ ПЛАСТИН С ШАРНИРНЫМ ОПИРАНИЕМ ПО КОНТУРУ ПРИ ИЗМЕНЕНИИ ЖЕСТКОСТИ СВЯЗЕЙ СДВИГА

The relationship between the fundamental frequency of free transverse vibrations ω of multilayer isotropic square plates in an unloaded state and their maximum deflections W 0 under the action of a uniformly distributed load depending on the number of layers and the rigidity of shear bonds is considered. The number of layers in a composite plate varies from two to ten. In the work, numerical studies of composite two-layer plates on compliant bonds were carried out using the finite element method. A finite element diagram of a multilayer plate is developed. The calculation was performed in the SCAD software package. The curves for different numbers of layers of multilayer plates “Deflection - stiffness of shear bonds” and “Frequency of transverse vibrations - stiffness of shear bonds” are plotted. It is shown that for composite multilayer square-shaped plates, the coefficient K coincides with an accuracy of 5.4% with the same coefficient for square plates of whole section.

Determination of frequency of free vibrations of hard foundation on the resilient founding under the action of impulse of explosion

Determination of frequency of free vibrations of hard foundation on the resilient founding under the action of impulse of explosion

Torsional vibration of cracked carbon nanotubes with torsional restraints using Eringen’s nonlocal differential model

Free torsional vibration of cracked carbon nanotubes with elastic torsional boundary conditions is studied. Eringen’s nonlocal elasticity theory is used in the analysis. Two similar rotation functions are represented by two Fourier sine series. A coefficient matrix including torsional springs and crack parameter is derived by using Stokes’ transformation and nonlocal boundary conditions. This useful coefficient matrix can be used to obtain the torsional vibration frequencies of cracked nanotubes with restrained boundary conditions. Free torsional vibration frequencies are calculated by using Fourier sine series and compared with the finite element method and analytical solutions available in the literature. The effects of various parameters such as crack parameter, geometry of nanotubes, and deformable boundary conditions are discussed in detail.

Finite-element solution of the problems of the theory of shells amenable to shear and compression

For the convenience of applying numerical methods, in particular the finite element method, to solving tasks of the theory of thin shells amenable to shear and compression, all key equations for the research the stress-strain state within the considered shells under static and dynamic loading, detecting proper frequencies of free vibrations, and the initial post-critical state, have been noted in the matrix form. Numerical schemes for solving the problems of statics and dynamics, stability and vibrations of flexible shells amenable to shear and compression have been constructed. Cite as: I. Ye. Bernakevych, P. P. Vahin, I. Ya. Koziy, “Finite-element solution of the problems of the theory of shells amenable to shear and compression,” Prykl. Probl. Mekh. Mat. , Issue 16, 98–106 (2018) (in Ukrainian), http://doi.org/10.15407/apmm2018.16.98-106