Free Vibration Frequencies Research Articles

Experiments of tuned liquid damper (TLD) on the reduced shear frame model under harmonic loads

In this study shaking table tests which are applied on 3-Storey reduced shear frame models with TLD’s subjected to harmonic loadings are presented. Firstly, free vibration experiments are conducted on the structure and 1st free vibration frequency of the structure is determined. The structure is shaken under harmonic loading at a frequency equal to 1st frequency of the structure which provides the resonance condition. Displacements and accelerations are measured at storey levels of the structure. A container in a rectangular prism shape is manufactured as a TLD model. Liquid is poured in the container and the same experiments are repeated at different liquid heights. The effect of TLD application on the structural models considering displacement and acceleration of the structure are investigated. In addition the effect of TLD application and its allocation at different storey levels are calculated experimentally. As a result of the conducted experiments, most convenient TLD models considering both displacement and acceleration behavior are determined. It is observed that all the damping models cause significant levels of reduction in seismic behavior of the structure under harmonic loading.

Buckling and dynamic characteristics of a laminated cylindrical panel under non-uniform thermal load

Buckling and free vibration behavior of a laminated cylindrical panel exposed to non-uniform thermal load is addressed in the present study. The approach comprises of three portions, in the first portion, heat transfer analysis is carried out to compute the non-uniform temperature fields, whereas second portion consists of static analysis wherein stress fields due to thermal load is obtained, and the last portion consists of buckling and prestressed modal analyzes to capture the critical buckling temperature as well as first five natural frequencies and associated mode shapes. Finite element is used to perform the numerical investigation. The detailed parametric study is carried out to analyze the effect of nature of temperature variation across the panel, laminate sequence and structural boundary constraints on the buckling and free vibration behavior. The relation between the buckling temperature of the panel under uniform temperature field and non-uniform temperature field is established using magnification factor. Among four cases considered in this study for position of heat sources, highest magnification factor is observed at the forefront curved edge of the panel where heat source is placed. It is also observed that thermal buckling strength and buckling mode shapes are highly sensitive to nature of temperature field and the effect is significant for the above-mentioned temperature field. Furthermore, it is also observed that the panel with antisymmetric laminate has better buckling strength. Free vibration frequencies and the associated mode shapes are significantly influenced by the non-uniform temperature variations.

Thermo-mechanical vibration of rotating axially functionally graded nonlocal Timoshenko beam

The free vibration analysis of rotating axially functionally graded nanobeams under an in-plane nonlinear thermal loading is provided for the first time in this paper. The formulations are based on Timoshenko beam theory through Hamilton’s principle. The small-scale effect has been considered using the nonlocal Eringen’s elasticity theory. Then, the governing equations are solved by generalized differential quadrature method. It is supposed that the thermal distribution is considered as nonlinear, material properties are temperature dependent, and the power-law form is the basis of the variation of the material properties through the axial of beam. Free vibration frequencies obtained are cantilever type of boundary conditions. Presented numerical results are validated by comparing the obtained results with the published results in the literature. The influences of the nonlocal small-scale parameter, angular velocity, hub radius, FG index and also thermal effects on the frequencies of the FG nanobeams are investigated in detail.

Experimental investigation on buckling and free vibration behavior of woven natural fiber fabric composite under axial compression

Influence of axial compression load on buckling and free vibration characteristics of natural fiber fabric polymer composite beam is analyzed experimentally. Critical buckling strength, free vibration frequencies and modal loss factors are obtained and analyzed. It is found that buckling strength increases with number of layers of fabric in composite. It is also observed that weaving pattern of the fabric influences buckling strength of the composite and basket type woven fabric enhances the buckling strength compared to plain and herringbone woven fabric composites. Sandwich composites with glass fiber fabric facing layer and natural fiber fabric as core layer having higher buckling strength. Free vibration frequency reduces with increase in axial compression load while modal damping factor increases in the pre-buckling region. However, this behavior reverses in the post-buckling region. The load-deflection obtained experimentally is compared with finite element result obtained considering the geometric non-linearity.

Badania parametrów dynamicznych i materiałowych elementów żelbetowych z dodatkiem zeolitu i plastyfikatora

Celem artykułu jest przedstawienie wyników badań parametrów tłumienia drgań oraz cech materiałowych konstrukcji żelbetowych z dodatkiem tufów zeolitowych i plastyfikatora. Część pierwsza artykułu opisuje właściwości zeolitów oraz wpływ jego dodatku na parametry betonu. Część główna artykułu przedstawia wyniki badań parametrów tłumienia drgań żelbetowych modeli płytowych. Modele posiadają zmodyfikowane spoiwo, w którym część cementu zastąpiono zeolitem. Wartości współczynników tłumienia drgań określano metodą kolokacyjną i uwzględniano tylko pierwszą częstość drgań własnych modeli. Zestawiono również wartości charakterystyk materiałowych zmodyfikowanego betonu takie jak moduły Younga i wartości współczynników Poissona.

Tolerance and asymptotic modelling of dynamic problems for thin microstructured transversally graded shells

The objects of consideration are thin linearly elastic Kirchhoff-Love-type open circular cylindrical shells having a functionally graded macrostructure and a tolerance-periodic microstructure in circumferential direction. The first aim of this contribution is to formulate and discuss a new non-asymptotic averaged model for the analysis of selected dynamic problems for these shells. As a tool of modelling we shall apply the tolerance averaging technique. Contrary to the starting exact shell equations with highly oscillating, non-continuous and tolerance-periodic coefficients, governing equations of the proposed tolerance model have continuous and slowly varying coefficients depending also on a cell size. Hence, an important advantage of this model is that it makes it possible to study the effect of a microstructure size on the global shell dynamics (the length-scale effect). The second aim is to derive and discuss a certain asymptotic model being independent of a microstructure size. It will be shown that in the framework of the tolerance model not only the fundamental lower, but also the new additional higher-order free vibration frequencies can be derived and analysed.

Study of vibration characteristics for orthotropic circular cylindrical shells using wave propagation approach and multivariate analysis

A study of free vibration of orthotropic circular cylindrical shells is presented. The vibration control equations of shells are based on Flugge classical thin shell theory. Wave approach is used in the analysis, in which the boundary conditions of shells can be simplified according to the associated beam. The free vibration frequencies of shells can be obtained from a frequency polynomial equation of order 6. The parametric analysis of the free vibration of orthotropic cylindrical shells is investigated using a statistical method. The effects of geometrical parameters and material characteristics upon frequencies are investigated here. Multivariate analysis (MVA) can be a useful tool for this parametric study. Some statistical characteristics, including correlation analysis and ANOVA are applied. ANOVA has been conducted to predict the statistical significance of the various factors. Calculations are performed in the Minitab statistical software. The results show that the L/R, h/R and m have larger effects on the lowest frequency. The importance of input parameters is ranked according to their contributions to the total variance. A knowledge and data visualization approach, Self-organizing mapping (SOM) is also adopted here for mining some intrinsic characteristics of shells.

Tolerance modelling of free vibration frequencies of thin functionally graded plates with one-directional microstructure

A tolerance modelling approach for problems of periodic media and structures is adopted here to investigate free vibrations of thin functionally graded plates with a microstructure along one direction in the planes parallel to the plate midplane. The size of the microstructure is assumed to be of an order of the plate thickness. To show the effect of the microstructure on dynamic behaviour of these plates the tolerance modelling method is used, cf. Jędrysiak (2013). This method leads to model equations with smooth functional coefficients, which describe the effect of the microstructure size of these plates not only in dynamic but also in stationary problems. In order to take into account this effect two models – the asymptotic-tolerance model and the tolerance model are used. As an example of applications of these models free vibration frequencies for simply supported rectangular plate are calculated. A certain verification for some cases is also shown by using the finite element method.

Axial dynamics of a nanorod embedded in an elastic medium using doublet mechanics

This study investigates the axial vibration of carbon nanotubes (CNTs) embedded in an elastic medium using scale dependent doublet mechanics (DM) theory. Governing equations and all boundary conditions of CNTs are derived based on the variational principle. Free vibration frequencies are obtained and compared with the classical elasticity results for clamped-clamped (C-C) and clamped-free (C-F) boundary conditions. The effect of elastic medium stiffness, nanorod length and doublet separation distance on the axial vibration is examined. It is obtained that important differences exist between vibration frequencies predicted by classical elasticity theory and DM. DM theory can be used in the nano length scale design of structures.

Modified von Kármán equations for elastic nanoplates with surface tension and surface elasticity

In this paper, modified von Kármán equations are derived for Kirchhoff nanoplates with surface tension and surface tension-induced residual stresses. The simplified Gurtin-Murdoch model which does not contain non-strain displacement gradients in surface stress-strain relations is adopted, so that the von Kármán strain-compatibility equation can be expressed in terms of the stress function and deflection. The modified von Kármán equations derived here are different than the existing related models especially for elastic plates with in-plane movable edges. Unlike the existing models which predict a surface tension-induced tensile pre-stress for an elastic plate with in-plane movable edges, the present model predicts that this tensile pre-stress is actually cancelled by the surface tension-induced residual compressive stress. Our this result is consistent with recent clarification on similar issue for cantilever beams with surface tension, which implies that the existing models have incorrectly predicted an invalid tensile pre-stress for an elastic plate with in-plane movable edges which leads to significant overestimation of postbuckling load and free vibration frequencies. In addition, our numerical examples indicated that surface stresses can moderately increase or decrease postbuckling load and free vibration frequency of Kirchhoff nanoplate with all in-plane movable edges, depending on the surface elasticity parameters and the geometrical dimensions of nanoplates.

Free vibration model and theoretical solution of the tympanic membrane

Myringoplasty is one of the routine surgeries in the treatment of tympanic membrane (TM) perforation. Since the gross anatomical structure of the middle ear cannot be simulated in clinical practice, the surgery is mainly performed by experience and expertise. Based on the mechanical properties of TM, four hypotheses are presented where TM is simplified as a sectorial annulus membrane. This paper proposes a free vibration model of TM whose natural frequency of free vibration and the analytical expressions of corresponding natural vibration mode are obtained by variables separation method and Bessel function. Compared with the ANSYS numerical results, it shows that natural frequency calculated by finite element (FE) method is slightly higher because of the increase of model stiffness by ignoring high-order quantity. Compared with the experimental data from volunteers, it shows that the first-order, second-order and third-order principal resonances in the test are the combined effects of multiple natural frequencies and natural vibration modes instead of the single one. The theoretical model deduced in this paper is in higher precision with comparatively fewer parameters. It provides more precise mechanical reference to myringoplasty by calculating the response of the normal human ear.

Analysis of thermoelastic response in functionally graded hollow sphere without load

ABSTRACTIn this article, a model concerning free vibrations of spherically symmetric, thermoelastic, isotropic, and functionally graded sphere has been developed and analyzed in the context of linear theory of generalized thermoelasticity with one relaxation time. Laplace transform has been used to solve the problem which yields natural frequencies of free vibrations without performing inversion of the transform. The analytical results for coupled, uncoupled, and homogeneous spheres have been deduced as special cases of the general case. Natural frequencies of first 10 modes of vibrations have been obtained for different values of grading index of cobalt material regarding coupled thermoelastic, generalized thermoelastic, and elastic functionally graded spheres. The frequency shift and thermoelastic damping for Fourier and non-Fourier processes of heat propagation, temperature change, radial, and hoop stresses have been presented graphically. It has been analyzed here that grading index parameter helps in detecting the strength of signals in such material devices and the thermal relaxation time contributes in improving the quality of signals. The analysis also leads to the fact that grading index parameter is useful from design point of view and it can be tailored to specific applications for controlling the stress.

The effect of finite strain on the nonlinear free vibration of a unidirectional composite Timoshenko beam using GDQM

In this manuscript, free vibrations of a unidirectional composite orthotropic Timoshenko beam based on finite strain have been studied. Using Green-Lagrange strain tensor and comprising all of the nonlinear terms of the tensor and also applying Hamilton's principle, equations of motion and boundary conditions of the beam are obtained. Using separation method in single-harmonic state, time and locative variables are separated from each other and finally, the equations of motion and boundary conditions are gained according to locative variable. To solve the equations, generalized differential quadrature method (GDQM) is applied and then, deflection and cross-section rotation of the beam in linear and nonlinear states are drawn and compared with each other. Also, frequencies of carbon/epoxy and glass/epoxy composite beams for different boundary conditions on the basis of the finite strain are calculated. The calculated frequencies of the nonlinear free vibration of the beam utilizing finite strain assumption for various geometries have been compared to von Karman one.

Simple estimation of effective elastic constants for thin plates with regular perforations and an application to the vibration of distillation column trays

The horizontal perforated sheet metal plates are commonly used in the process industries as trays in distillation columns, important internal parts for fractionating the input liquid mixture. Normally, the operating performance of such trays is satisfactory. However, cases have been reported of abnormally high levels of tray vibration during operation at particular conditions. The trays then experienced fatigue cracking accompanied by the loosening of bolts and fixings, which led to expensive failures. The excitation of structural resonance was suspected as a component in flow-induced vibration. Using linear stress superposition, a simple but robust analytical method is developed to provide high-quality predictions for the stress and strain distributions for in-plane loaded thin perforated plates with periodic hole arrangements. This approach is built on the classical solution for the elastic stress field around a single circular hole in a large plate. The perforated plates with square penetration patterns are investigated in this article, although the same approach is applicable to any regular penetration pattern. Stress concentration factors as well as the effective elastic constants, which can be used to describe the bending properties of the perforated plates, are then verified against both the established theoretical solutions and the results from finite element simulations. Excellent agreement to both previously published physical experiments and complex modelling is observed in all cases, with small-to-medium (up to 40%) hole-area fraction. The proposed analytical method is much simpler and computationally efficient than finite element analysis. The computed effective elastic constants are used in a finite element modal analysis to estimate the free vibration frequencies of a stiffened distillation column tray example; the first 30 vibration modes are found to be almost uniformly distributed between 25 and 70 Hz, which matches the vibration frequency range reported from plant operations.

Mathematical model of the dynamics of micropolar elastic thin beams. Free and forced vibrations

A method of hypotheses has been developed to construct a mathematical model of micropolar elastic thin beams. The method is based on the asymptotic properties of the solution ofan initial boundary value problem in a thin rectangle within the micropolar theory of elasticity with independent displacement and rotation fields. An applied model of the dynamics of micropolar elastic thin beams was constructed in which transverse shear strains and related strains are taken into account. The constructed dynamics model was used to solve problems of free and forced vibrations of a micropolar beam. Free vibration frequencies and modes, forced vibration amplitudes, and resonance conditions were determined. The obtained numerical calculation results show the specific features of free vibrations of thin beams. Micropolar thin beams have a free vibration frequency which is almost independent of the thin beam size, but depends only on the physical and inertial properties of the micropolar material. It is shown for the micropolar material that the free vibration frequency values of beams can be readily adjusted and hence a large vibration frequency separation can be achieved, which is important for studying resonance.

Free vibration analysis of axially functionally graded linearly taper beam on elastic foundation

In the present study non-linear free vibration analysis is performed on a linearly tapered Axially Functionally Graded (AFG) beam resting on an elastic foundation with different boundary conditions. Firstly the static problem is carried out through an iterative scheme using a relaxation parameter and later on the subsequent dynamic problem is solved as a standard eigen value problem. Minimum potential energy principle is used for the formulation of the static problem and for the dynamic problem Hamilton's principle is utilized. The free vibrational frequencies are tabulated for different taper parameter and different foundation stiffness. The dynamic behaviour of the system is presented in the form of backbone curves in dimensionless frequency-amplitude plane.

Free vibrations of circular cylindrical shells with a small added concentrated mass

The effect of a small added mass on the frequency and shape of free vibrations of a thin shell is studied using shallow shell theory. The proposed mathematical model assumes that mass asymmetry even in a linear formulation leads to coupled radial flexural vibrations. The interaction of shape-generating waves is studied using modal equations obtained by the Bubnov–Galerkin method. Splitting of the flexural frequency spectrum is found, which is caused not only by the added mass but also by the wave-formation parameters of the shell. The ranges of the relative lengths and shell thicknesses are determined in which the interaction of flexural and radial vibrations can be neglected.

Generalized Timoshenko–Reissner model for a multilayer plate

A multilayer plate with isotropic (or transversally isotropic) layers strongly differing in rigidity is considered. This plate is reduced to an equivalent homogeneous transversally isotropic Timoshenko–Reissner plate whose deflections and free transverse vibration frequencies are close to those of the multilayer plate. By comparison with the exact solution of test three-dimensional problems of elasticity, the error of the proposed method is estimated both for the static problem and for free vibrations. This comparison can readily be carried out for the hinged edges of the plate, and explicit approximate formulas are obtained for the vibration frequencies. The scope of the proposed model turned out to be rather wide (the Young moduli of soft and rigid layers can differ by a factor of 1000). In the case of boundary conditions other than hinged support, a closed-form solution cannot be constructed in general. For rigidly fixed edges, the asymptotic method proposed by V. V. Bolotin is generalized to the case of a Timoshenko–Reissner plate.

Identification of the Elastic and Damping Characteristics of Soft Materials Based on the Analysis of Damped Flexural Vibrations of Test Specimens

A method for identification of the elastic and damping properties of soft materials in tension–compression and shear in a composition with a rigid (base) material is proposed. The method is based on the analysis of the lowest free vibration frequencies and amplitude dependences of internal damping parameters of three-layer test specimens of an appropriate structure: the base material is placed in the middle of the specimens for tension–compression, but as outer layers for shear. Identification of the elastic properties in tension−compression and shear is based on a comparison of calculated and experimental frequencies of corresponding test specimens. It is shown that the elastic properties found for a technical rubber depends on vibration frequencies of the test specimen. A quadratic functional containing experimental and calculated parameters of internal damping of test specimens is employed to identify the damping properties of soft materials. A three-layer finite element developed for this purpose is proposed for a numerical evaluation of the design parameters of internal damping of a three-layer test specimen. Calculations were carried out to identify the damping characteristics of the technical rubber in tension–compression and shear from the parameters of internal damping of test specimens with a D16AT aluminum alloy as the base material.

Natural Frequency and Mode Shapes of Exponential Tapered AFG Beams on Elastic Foundation

A displacement based semi-analytical method is utilized to study non-linear free vibration and mode shapes of an exponential tapered axially functionally graded (AFG) beam resting on an elastic foundation. In the present study geometric nonlinearity induced through large displacement is taken care of by non-linear strain-displacement relations. The beam is considered to be slender to neglect the rotary inertia and shear deformation effects. In the present paper at first the static problem is solved through an iterative scheme using a relaxation parameter and later on the subsequent dynamic analysis is carried out as a standard eigen value problem. Energy principles are used for the formulation of both the problems. The static problem is solved by using minimum potential energy principle whereas in case of dynamic problem Hamilton’s principle is employed. The free vibrational frequencies are tabulated for exponential taper profile subject to various boundary conditions and foundation stiffness. The dynamic behaviour of the system is presented in the form of backbone curves in dimensionless frequency-amplitude plane and in some particular case the mode shape results are furnished.