Free Vibration Frequencies Research Articles

Size‐dependent Levinson beam theory for thermal vibration of a nanobeam with deformable boundary conditions

AbstractIn this study, an eigen‐value problem for deformable boundary conditions in nonlocal elasticity is described using Levinson beam theory. Firstly, the nanobeam has been modeled by placing two springs that can be deformed in the downward direction. These springs control the amount of downward displacement at the ends. In the analytical solution, the displacement points are defined by two coefficients and the interior part of the nanobeam deflection is expressed by Fourier sine series. Stokes’ transformation is preferred to enforce the boundary conditions to the desired point. After the mathematical operations, a matrix of coefficients including the general elastic spring constants has been found. The eigenvalues of this coefficient matrix give the frequencies of the Levinson nanobeam. The effect of some parameters on the free vibration frequencies is shown in a series of graphs and tables.

Determination of the natural frequencies of the above-ground sections of pipelines transporting an incompressible fluid

Under certain conditions, above-ground sections of pipelines are subject to vibration. Increasing the vibration amplitudes beyond the limits can cause the object to fail. In order to assess the vibration resistance of pipelines and to find the most effective ways of detuning from the resonant frequencies of disturbances, it is necessary to determine the natural frequencies of oscillations and the characteristics of the oscillation processes. The authors of the article developed a mathematical model to determine the frequency of free vibrations of above-ground pipeline sections, taking into account the influence of internal pressure and the velocity of the pumped fluid. As a result, a numerical solution based on the Bubnov-Galerkin method was obtained and a sufficient number of series elements in the approximate solution was determined. The authors performed numerical experiments. These showed that the frequencies should be determined for each section under specific conditions according to the pumping mode.

Effect of Graphene Reinforcement on Free Vibration and Material Properties of the FG-GPLRC Porous Cantilever Torsional Plate

This paper researches the free vibrations and corresponding material parameters of a functional gradient graphene platelets-reinforced composite (FG-GPLRC) cantilever torsional plates with both pore and graphene variations in the transverse direction. Utilizing the feature of closed-cell cellular solids, the mixture rule, and the modified Halpin–Tsai model, the material parameters of the composite are determined for different volume fractions of component materials. Then, using the classical plate theory (CLPT), the Rayleigh–Ritz technique and polynomials, the dynamic equation that can be used to obtain the free vibration mode shapes and frequencies of the rotating cantilever torsional plate is given. Comparison studies with the previous calculation results from available literature and the finite element (FE) models of cantilever plates are conducted, and the correctness of the present theoretical formulation and numerical calculation is verified. Finally, the effects of graphene platelet (GPL) distribution, porosity distribution (PD), GPL content, rotational speed, and average geometric size of GPL on free vibrations of the system are studied in depth.

Free vibrations of rectangular plates

The paper investigates free vibrations of rectangular metal plates. The finite element method and analytical calculation are particularly used to determine the vibration frequency. The analytical calculation is based on the equation of motion of a thin rectangular plate. The asymptotic method is applied to determine the dynamic edge effect. As a result, the free vibration frequency is determined for the rectangular metal plate. The finite element analysis is performed in Lira and SolidWorks software packages. For this, a solid plate model with sensors is created to measure the free vibration frequency; the effective mass participation factor was determined.The plate vibration tests were conducted to confirm the results of analytical calculations. The method of smooth sinusoidal vibrations is used. Spectrum graphs of the plate vibrations are suggested based on the vibration acceleration of sensors. Errors are identified in the free vibration frequencies depending on the applied method. The paper does not consider frequencies with the low effective mass participation factor.

Algorithm for simulation of vibration effects during track tests of aircraft and missile technology

Track high-speed ground tests of products of aviation and rocket technology make it possible to experimentally confirm the aerodynamic and strength characteristics of new aircraft or their components. Experimental installation Rocket rail track 3500, located on the territory of the Federal State Enterprise Scientific Test Range of Aviation Systems named after L. K. Safronov (FKP GkNIPAS named after L.K. Safronov), is a complex hydro-gas-dynamic engineering structure. The main part of this installation is a rail track placed on a special concrete base, which is based on piles to eliminate the influence of viscoelastic ground vibrations. Between the rails, taking into account the profile of the track, a hydrodynamic multi-level tray is made, designed for hydraulic braking of the stored track sled. The movable rocket track sled rests on slip shoes covering the top rail head. On the track sleigh, solid propellant rocket engines and the object of study are placed. The article describes the approach to the calculation of determining the dynamic loads acting on the elastic structure of the track sled with the test object. The design is represented by a schematic spatial model of elastic beams in the form of rods, plates, pipes with equivalent mass and stiffness, interconnected by elastic links. A model of non-stationary aerodynamic forces is formulated taking into account viscosity for numerical calculations of the flow around the sled structure. Programs have been developed and modeling of the aerodynamic interaction in the case of a supersonic air flow around the structure of a rocket sled has been carried out. The results of vibrational accelerations of elements, components of the sled and the test object, depending on the speed of movement, were obtained by calculation. The forms and frequencies of natural free vibrations of the sled structure are determined, and the densities of the vibration acceleration spectra are calculated.

Free Vibration Response of a Steel Liquid Storage Tank with Porous and Perforated Columns via an Exact Continuum Method

The presence of predictable cavities inside a structure is often preferable as it will reduce the vibration amplitude. In this article, inspired by the traditional mass attached to the beam system, in order to solve the problem of free vibration of the steel liquid storage tank with a column made of porous and perforated materials using an exact analytical continuum model is formed. The free vibration response of the mass attached column made of porous materials and holes is established by using a new analytical method known as the Fourier series with Stokes’ transform. The free vibration frequencies are obtained by using an eigen-value approximation, and the influences of the number of holes, filling ratio, porosity and other parameters on the free vibration response are explored. It is shown that an increase in the mass parameter, filling ratio and porosity parameter of the column with an attached mass system could significantly affect the free vibration response of the system.

Numerical Evaluation of Frequencies and the Modes of Free Vibrations of Isosceles Triangular Plates with Free Edges

Free vibrations are analyzed for isotropic plates in the form of isosceles triangles. We deduce the formula for the evaluation of the frequencies of free vibrations for plates of regular triangular shape with free edges and compute the coefficients of vibration mode and boundary conditions. The frequencies and modes of free vibrations of the isotropic thin isosceles triangular plates with free edges and different apex angles are computed by the finite-element method.

Comparison of Free Vibrations Frequencies of a Laminated Composite Honeycomb Sandwich Plate for Different Boundary Conditions and Different Lamination Angles

Tabakalı kompozit sandviç elemanlar, inşaat, otomotiv, havacılık ve denizcilik gibi birçok sektörde yoğun bir şekilde kullanılmaktadır. Yüksek mukavemet/ağırlık oranı, aşındırıcı olmayan, iyi yüzey vb gibi avantajlı özelliklere sahip olmalarından dolayı bu malzemelerin kullanımı yoğun olarak tercih edilmekte ve bu malzemelerle ilgili yapılan çalışmalar günden güne artmaktadır. Bu çalışmada, tabakalı kompozit sandviç plağın modal analizi yapılmıştır. Analizlerde, sonlu elemanlar modeli, ANSYS ve ABAQUS programları kullanılarak oluşturulmuştur. Tabakalı kompozit sandviç plağın çeşitli tabakalanma durumlarında serbest titreşim analizleri yapılarak doğal frekanslar elde edilmiştir. Elyaf açılarının, tabaka dizilim sırasının ve sınır koşullarının tabakalı kompozit balpeteği sandviç plağın doğal frekansları ve mod şekilleri üzerinde etkili olduğu görülmektedir.

Free vibration analysis of Timoshenko pipes with fixed boundary conditions conveying high velocity fluid

Sever vibration will be induced due to the high flow velocity in the pipe. When the flow velocity exceeds the critical value, the static equilibrium configuration of the pipe loses its stability, and the vibration properties change accordingly. In this paper, the free vibration characteristics of the pipe with fixed-fixed ends are revealed in the supercritical regime. Based on the Timoshenko beam theory, the governing equations of the nonlinear vibration near the non-trivial static equilibrium configuration are established. The influences of system parameters on equilibrium configuration, critical velocity, and free vibration frequency is analyzed. The effects of supercritical velocity in different ranges on the natural frequencies are revealed. In addition, the comparison with the Euler-Bernoulli pipe model shows that the differences in critical velocity, equilibrium configuration, and frequency are still significant even the length-diameter ratio is large. The increase of the flow velocity reduces the difference of non-trivial static equilibrium configurations, but eventually aggravates the difference of natural frequencies. Within a certain supercritical velocity range, the vibration difference between the two pipe models is small, beyond this range, the vibration difference increases significantly.

Geometrically Nonlinear Aeroelastic Flutter Characteristic of Laminated Composite Shell Panels Under Supersonic Flow

The nonlinear aeroelastic flutter characteristics of laminated composite curved (flat and cylindrical) panels are reported in this paper by obtaining the finite element (FE) solutions under the supersonic flow. The panel model has been derived from the third-order shear deformation theory framework including full geometrical nonlinearity via Green–Lagrange strain. In contrast, the flutter due to aerodynamic loading is included via the first-order piston theory. The current predicted solution accuracy and their validations have been demonstrated by relating the free vibration frequency, coalescence frequency, and critical aerodynamic pressure with the available numerical data. The initial free vibration eigenvalue responses are compared with in-house experimental values. Finally, a few numerical examples are presented by varying parameters like the effect of fiber orientation, flow angle, end boundary conditions, aspect ratio, modular ratio, thickness ratio, and amplitude ratio on the supersonic flutter boundaries of shell panels. The critical aerodyanamic pressure for simply supported cross-ply flat panels increases by 42.85% when the amplitude ratio increases from 0 to 0.75.

INSTRUMENT FOR MEASURING PULSATION CHARACTERISTICS IN TURBULENT FLOWS

The paper considers the design of a two-component strain gauge construction for measuring pulsation characteristics in turbulent flows.
 The analysis is carried out and the shortcomings of the existing sensor models and their influence on the reliability and accuracy of the experimental results are shown. The limited frequency range of such sensors is also emphasized.
 The design of the previous types of sensors is based on a plate on which wire or film strain gauges are glued. In such models, the main part of the perceiving hydrodynamic pressure goes to the formation of stresses in the elastic plate, and only a small part (up to 4%) - to create stresses in the strain-sensitive elements. Such sensors require an increase in the size of sensitive elements, this causes an additional disturbance in the flow, which in turn also reduces the accuracy of the measurement.
 The use of glue in the design of the sensor is associated with adverse effects such as creep and aging of the glue.
 The proposed sensor used strain gauges in the form of constantan wires stretched parallel to each other, which helped to get rid of elastic plates and glue, and also reduced the size of the sensing element.
 A construction of the sensor is shown and its design is described.
 The sensor has a free vibration frequency in water of more than 300 Hz, which is more than an order of magnitude higher than the frequency of water flow oscillations. This allows a more accurate measurement of pulsation characteristics.
 The principle of operation of the strain gauge is considered.
 The presence in the construction of four correspondingly located tensile strands made it possible to measure the longitudinal and transverse components of the pulsation velocities in turbulent flows.

Buckling and free vibrations behaviour through differential quadrature method for foamed composites

The current work focuses on predicting the buckling and free vibration frequencies (fn) of cenosphere reinforced epoxy based syntactic foam beam under varying loads. Critical buckling loads (Ncr) and fn are predicted using the differential quadrature method (DQM). Ncr and fn have been calculated for beams of varying cenosphere volume fractions subjected to axial load under clamped-clamped (CC), clamped-simply (CS), simply-simply (SS), and clamped-free (CF) boundary conditions (BC′s). Upon increasing the cenosphere volume fraction, Ncr and fn of syntactic foam composites increases. These numerical outcomes are compared with the theoretical values evaluated through the Euler-Bernoulli hypothesis and further compared with experimental outcomes. Results are observed to be in precise agreement. The results of the DQM numerical analysis are given out for the different BC′s, aspect ratios, cenosphere volume fractions, and varying loads. It is perceived that depending on the BC′s, the type of axial varying loads and aspect ratios has a substantial effect on the Ncr and fn behaviour of the syntactic foam beams. A comparative study of the obtained results showed that the beam subjected to parabolic load under CC boundary conditions exhibited a higher buckling load.

Влияние клеевых соединений после термоциклических испытаний на прочностные характеристики и частоту свободных колебаний

The results of shear tests of adhesive joints of a polymer composite material (PCM) are considered, taking into account the operational factors and manufacturing technology of composite materials. The results of vibration tests and computer modelingto determine free vibration frequencies (FVF) are presented. Installations for thermal cycling and vibration testing, as well as the test modes, are described

On the modeling of boundary conditions for determining the critical speeds of rotation of rotors

The article is devoted to the definition of boundary conditions that allow achieving adequate results when calculating the critical speeds of rotation of rotors based on finite element models. Experimental determination of the frequencies of free vibrations of a laboratory rotor on ball bearings was carried out using a vibration measuring complex developed by the authors. Numerical studies of the free vibrations of a laboratory rotor and a cylindrical pipe with a wall thickness to length ratio of 1/10 were performed on the basis of a three-dimensional finite element model under various boundary conditions. Movable and fixed cylindrical hinges corresponding to floating bearings are used as supports in the calculation scheme of the laboratory rotor. Due to the fact that the dimensions of the bearings along the length of the shaft are small compared to the total length, it is assumed in the calculation model that the shaft is hinged along the edges at separate points of one circle on each bearing. Shaft mounting options at two, four and eight points are considered. The options for fastening a cylindrical pipe in the extreme sections at two points, a quarter of a circle and a full circle are considered. Comparison of experimental and numerical results, solutions obtained analytically and on the basis of the finite element method is carried out. An analysis of the modes of natural vibrations showed that with various options for fixing a cylindrical pipe, vibration modes arise associated with the deformation of the pipe as a shell. For example, when fixing a pipe along a quarter of a circle, among the forms №1-№10, only the second corresponds to the deformation of the pipe as a beam. The results of the study of free vibrations of a laboratory rotor show that the best option for boundary conditions, which makes it possible to approach the results of the experiment, is fastening on each bearing at two points located on the neutral line of the cross section when the rotor is bent. Numerical studies of the eigen frequencies of a cylindrical pipe show that it is with this type of boundary conditions that one can obtain results that are closest to the analytical solution.

Finite element mesh refinement for in-plane and out-of-plane vibration of variable geometrical Timoshenko beams based on superconvergent vibration modes

PurposeIn this paper, a superconvergent patch recovery method is proposed for superconvergent solutions of modes in the finite element post-processing stage of variable geometrical Timoshenko beams. The proposed superconvergent patch recovery method improves the solution speed and accuracy of the finite element analysis of a curved beam. The free vibration and natural frequency of the beam were considered for studying forced vibrations and structural resonance. Beam vibration mode analysis was performed for high-precision vibration mode solutions and frequency values. The proposed method can be used to compute beam vibration modes of beams with different shapes and boundary conditions as well as variable cross sections and curvatures. The purpose of this paper is to address these issues.Design/methodology/approachAn adaptive method was proposed to analyse the in-plane and out-of-plane free vibrations of the variable geometrical Timoshenko beams. In the post-processing stage of the displacement-based finite element method, the superconvergent patch recovery method and high-order shape function interpolation technique were used to obtain the superconvergent solution of mode (displacement). The superconvergent solution of mode was used to estimate the error of the finite element solution of mode in the energy form under the current mesh. Furthermore, an adaptive mesh refinement was proposed by mesh subdivision to derive an optimised mesh and accurate finite element solution to meet the preset error tolerance.FindingsThe results computed using the proposed algorithm were in good agreement with those computed using other high-precision algorithms, thus validating the accuracy of the proposed algorithm for beam analysis. The numerical analysis of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams helped verify that the solutions of frequencies were consistent with the results obtained using other specially developed methods. The proposed method is well suited for the mesh refinement analysis of a curved beam structure for analysing the changes in high-order vibration mode. The parts where the vibration mode changed significantly were locally densified; a relatively fine mesh division was adopted that validated the reliability of the mesh optimisation processing of the proposed algorithm.Originality/valueThe proposed algorithm can obtain high-precision vibration solutions of variable geometrical Timoshenko beams based on more optimized and reasonable meshes than the conventional finite element method. Furthermore, it can be used for vibration problems of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams. The proposed algorithm can be extended for application in superconvergent computation and adaptive analysis of finite element solutions of general structures and solid deformation fields and used for adaptive analysis of more complex plates, shells and three-dimensional structures. Additionally, this method can analyse the vibration and stability of curved members with crack damage to obtain high-precision vibration modes and instability modes under damage defects.

Variable stiffness optimization algorithm for vibration optimization of variable-stiffness composite plates

A novel variable stiffness optimization (VSO) algorithm is proposed for vibration optimization of variable-stiffness composite (VSC) plates uisng linear variation fiber path function. Firstly, an independent 2D-sampling optimization method (2DSOind) is employed to generate a good initial point. Then, optimization is conducted in multiple phases, within each phase, the problem is simplified to an approximately one-dimensional linear search problem. By dividing the fiber angles into two groups, VSO algorithm designs the laminate between two fiber angle groups sequentially and iteratively, making the stiffness of VSC plate stiffened part by part. Lastly, the VSC plate is redesigned accounting for the curvature constraint. The sequential permutation search (SPS) algorithm is coupled into VSO to optimize one group of fiber angles whilst the angles of another group are fixed during the iteration. Under a variety of boundary conditions, the free vibration frequencies of VSC plates are optimized using VSO. In comparison to conventional layerwise optimization (LO) and genetic algorithm (GA), the present results demonstrate much higher efficiency and similar robustness. Moreover, the maximum fundamental frequency of VSC plates can be improved by about 10% in comparison to straight fiber laminates.

Extended tolerance modelling of dynamic problems for thin uniperiodic cylindrical shells

Dynamic problems of thin linearly elastic Kirchhoff–Love-type circular cylindrical shells having geometrical, elastic and inertial properties densely and periodically varying in circumferential direction (uniperiodic shells) are studied. In order to take into account the effect of a cell size on the global dynamic behaviour of such shells (the length-scale effect), a new mathematical averaged non-asymptotic model is formulated. This so-called the general tolerance model is derived by applying a certain extended version of the well-known tolerance modelling technique. Governing equations of this averaged model have constant coefficients depending also on a microstructure size, contrary to the starting exact shell equations with periodic, non-continuous and highly oscillating coefficients (the well-known governing equations of linear Kirchhoff–Love theory of thin elastic cylindrical shells). The effect of a cell size on the transversal free vibrations of an uniperiodic shell strip is studied. It will be shown that within this general tolerance model not only fundamental cell-independent, but also the new additional cell-dependent free vibration frequencies can be derived and analysed. The obtained results will be compared with the corresponding results derived from the knownnon-asymptotic standard tolerance model and from the asymptotic one.

A hybrid averaging and harmonic balance method for weakly nonlinear asymmetric resonators

Models for nonlinear vibrations commonly employ polynomial terms that arise from series expansions about an equilibrium point. The analysis of symmetric systems with cubic stiffness terms is very common, and the inclusion of asymmetric quadratic terms is known to modify the effective cubic nonlinearity in weakly nonlinear systems. When using low (second, in this case)-order perturbation methods, the net effect in these cases is found to be a monotonic dependence of the free vibration frequency on the amplitude squared, with a single term that depends on the coefficients of the quadratic and cubic terms. However, in many applications, such a monotonic dependence is not observed, necessitating the use of techniques for strongly nonlinear systems, or the inclusion of higher-order terms and perturbation methods in weakly nonlinear formulations. In either case, the analysis involves very tedious and/or numerical approaches for determining the system response. In the present work, we propose a method that is a hybrid of the methods of averaging and harmonic balance, which provides, with relatively straightforward calculations, good approximations for the free and forced vibration response of weakly nonlinear asymmetric systems. For free vibration, it captures the correct amplitude–frequency dependence, including cases of non-monoticity. The method can also be used to determine the steady-state response of damped, harmonically driven vibrations, including information about stability. The method is described, and general results are obtained for an asymmetric system with up to quintic nonlinear terms. The results are applied to a numerical example and validated using simulations. This approach will be useful for analyzing a variety of system models with polynomial nonlinearities.

Exact Frequencies for Free Vibration of Exponential and Polynomial AFG Beams with Lumped End Masses and Elastic Supports

Exact Frequencies for Free Vibration of Exponential and Polynomial AFG Beams with Lumped End Masses and Elastic Supports

Adaptive mesh refinement for finite element analysis of the free vibration disturbance of cylindrical shells due to circumferential micro-crack damage

PurposeModerately thick circular cylindrical shells are widely used as supporting structures or storage cavities in structural engineering, rock engineering, and aerospace engineering. In practical engineering, shells often work with micro-cracks or defects. The existence of micro-crack damage may result in the disturbance of dynamic behaviours and even induce accidental dynamic disasters. The free vibration frequency and mode are important parameters for the dynamic performance and damage identification analysis. In particular, stiffness weakening of the local damage region leads to significant changes in the vibration mode, which makes it difficult for the mesh generated in the conventional finite element method to capture a high-precision solution of the local oscillation.Design/methodology/approachIn response to the above problems, this study developed an adaptive finite element method and a crack damage characterisation method for moderately thick circular cylindrical shells. By introducing the inverse power iteration method, error estimation, and mesh subdivision refinement technique for the analysis of finite element eigenvalue problems, an adaptive computation scheme was constructed for the free vibration problem of moderately thick circular cylindrical shells with circumferential crack damage.FindingsBased on typical numerical examples, the established adaptive finite element solution for the free vibration of moderately thick circular cylindrical shells demonstrated its suitability for solving the high-precision free vibration frequency and mode of cylindrical shell structures. The any order frequency and mode shape of cracked cylindrical shells under the conditions of different ring wave numbers, crack locations, crack depths, and multiple cracks were successfully solved. The influences of the location, depth, and number of cracks on the disturbance of dynamic behaviours were analysed.Originality/valueThis study can be used as a reference for the adaptive finite element solution of free vibration of moderately thick circular cylindrical shells with cracks and lays the foundation for further development of a high-performance computation method suitable for the dynamic disturbance and damage identification analysis of general cracked structures.