Natural Frequencies Of Flexural Vibration Research Articles

Free flexural vibrations of standard wide-flange H-beams with consideration of the shear effect

The subject of the paper is a simply supported standard wide-flange H-beam. Cross sections of this beam is analytically described as a three-layer structure. The shear effect in its succesive layers is taking into account with consideration of the classical shear stress formula called Zhuravsky shear stress. Based on Hamilton’s principle, two differential equations of motion are obtained. These equations are analytically solved and the fundamental natural frequency of flexural vibration for this beam is derived. Examplary calculations are carried out for selected five I-beams.

Determination of the Mass-Flow Rate of Blood in a Blood Vessel Using Natural Frequencies of Flexural Vibrations

The natural flexural vibrations of a blood vessel with moving blood are investigated. The vessel is subjected to the action of tensile force and pressure in it. The forces of inertia of the blood vessel, as well as the Coriolis and centrifugal forces induced by the motion of blood, are taken into account. The wave numbers are determined, and the frequency equation is found using the boundary conditions. Two frequencies of flexural vibrations can determine the speed parameter, the relative mass of blood per unit length of the blood vessel, and, as a consequence, the mass-flow rate of blood through the blood vessel. The results obtained can be used for the acoustic method of determining the blood speed, the relative blood mass per unit length of the blood vessel, and the mass-flow rate of blood through it.

Flexural vibrations of the pipeline on elastic supports with moving fluid

In the work investigated the flexural vibrations of the pipeline. Parts of the pipeline on both sides of the sagging section have elastic supports. It is assumed that a constant longitudinal force acts along the neutral line. An incompressible fluid flows through the pipe at a constant average speed. The influence of internal pressure in the pipe on these oscillations is taken into account. The direct problem of determining the eigenfrequencies of flexural vibrations of the pipeline by the Kirchhoff model using Ferrari formulas is solved. The frequency spectrum is determined depending on the fluid pressure, the elasticity of the supports, the velocity of the fluid through the pipe. Particular and limiting cases are considered, for example, when the stiffness of the supports is very large and when they are very small. Graphs of the dependence of the first and second eigenfrequencies on the velocity of the transported liquid at different values of the liquid density parameter are constructed. It is shown that with the growth of the velocity parameter there is a decrease in the natural frequencies of flexural vibrations of the pipeline, and the faster the higher the density parameter of the liquid. It is determined that with an increase in the mass of the liquid per unit length of the pipeline there is a decrease in the natural frequencies of bending vibrations of the pipe. It is found that with the increase in the mass flow through the pipe, the natural frequencies of bending oscillations also decrease. It is confirmed that the frequencies of flexural vibrations of the pipeline are the same for the cases of pipe fastening “rigid fixing — rigid fixing” and “free end — free end”. The results of the study will contribute to the development of methods of acoustic diagnostics and non-destructive testing and can find technical application for monitoring and diagnostics of pipeline systems.

Determining the velocity of a moving three-layered plate and the thickness of its filler using natural frequencies of flexural vibrations

Research has been performed on natural transverse vibrations of a portion of a constant length in a straight threelayered plate with a filler moving along the neutral line of a non-deformed state. The movement takes place between two rigidly fixed coaxial guides (clamps), the distance between them equalling the length of the vibrating portion. The longitudinal force is assumed to constantly act along the neutral line. It has been found that an increase in the natural frequencies of flexural vibrations of the plate occurs with an increase in the thickness of the filler. Using two frequencies of flexural vibrations, we can determine the velocity of the moving three-layered plate and the thickness of its filler. The results of the research work can find technological use in the problems on dynamics and strength of machines and mechanisms in manufacturing three-layered plate with fillers and can be used to determine the velocity of a plate and the thickness of its filler using two frequencies of flexural vibrations.

Flexural natural vibration characteristics of composite beam considering shear deformation and interface slip

Based on Hamilton's principle, the flexural vibration differential equations and boundary conditions of the steel-concrete composite beam (SCCB) with comprehensive consideration of the influences of the shear deformation, interface slip and longitudinal inertia of motion were derived. The analytical natural frequencies of flexural vibration were compared with available results previously observed by the experiments, the results calculated by the FE model and the other similar beam theories available in the open literatures. The comparison results showed that, the calculation results of the analytical and Timoshenko models had a good agreement with the results of the experimental test and FE model. Finally, the influences of shear deformation and interface slip on the flexural natural frequencies of the SCCB were discussed. The shear deformation effect increases with the increase of the mode orders of flexural natural vibration, and the flexural natural frequencies of the higher mode orders ignoring the influence of shear deformations effect would be overestimated. The interface slip effect decrease with the increase of the mode orders of flexural natural vibration, and the influence of the interface slip effect on flexural natural frequencies of the low mode orders is significant. The influence of the degree of shear connection on shear deformation effect is insignificant, and the low order modes of flexural natural vibration are mainly composed of the rotational displacement of cross sections.

Determining the velocity of a moving rod and the thickness of its coating using natural frequencies of flexural vibrations

Research has been performed on natural transverse vibrations of a portion of a constant length in a straight thin coated rod moving along the neutral line of a non-deformed state. The movement takes place between two rigidly fixed coaxial guides (clamps), the distance between them equalling the length of the rod moving portion. The longitudinal force is assumed to constantly act along the neutral line. It has been found that a decrease in the natural frequencies of flexural vibrations of the rod occurs with an increase in the velocity parameter. It is also shown that a decrease in the natural frequencies of flexural vibrations of the rod occurs with an increase of the coating thickness. Using two frequencies of flexural vibrations, we can determine the velocity parameter of the moving rod and the thickness of its coating. The results of the research work can find technological use in the problems on dynamics and strength of machines and mechanisms in manufacturing the coated products: in textile industry, wire manufacturing process, metallurgy (especially for rolling metal bars and strips), wire drawing, plastic products and paper rolls manufacturing and can be used to determine the velocity and the coating thickness of a moving rod, strip or wire using two frequencies of flexural vibrations.

SCALING DETECTION IN A PIPELINE WITH FLUID UNDER PRESSURE USING NATURAL FREQUENCIES OF FLEXURAL VIBRATIONS

Pipelines can form scale deposits on their walls. Erosion-corrosion processes in pipelines are characterized by the fact that deposition of corrosion products in some of their portions occurs simultaneously with wall thinning in the others. In this paper consideration is given to natural frequencies of flexural vibrations in a pipeline clamped at both ends and subjected to the action of tension force. The pipeline has wall scale deposits and is filled with fluid under pressure. The scale deposits are supposed to be uniformly distributed on the inner surface of the pipeline. Here, the Kirchhoff’s equation for flexural vibrations is applied. The frequency equation has been derived from the equation that dictates the type of flexural vibrations and also the boundary conditions for a pipeline clamped at its ends. Solutions have been found to the primal problem to determine the first and the second natural frequencies of flexural vibrations in the pipeline for the given scale density in the pipeline, axial force, wall thickness, inside pressure of the pipeline and uniform scale thickness on the pipe wall as well as to the inverse problem to determine scale density on the pipe wall and deposition thickness using two natural frequencies of flexural vibrations in the pipeline. It has been found that an increase in the density of scale deposits on the pipe wall results in smaller natural frequencies of pipe flexural vibrations. Besides, the research has shown that a reduction in the pipe inside area or an increase in the scale thickness results in smaller natural frequencies of pipe flexural vibrations. Using two natural frequencies of pipe flexural vibrations we can determine the density of scale deposits on the pipe wall and pipe inside area or scale thickness. The investigation outcomes can be applied to assess axial loading and pipe wall thickness using two natural frequencies of flexural vibrations.

A Large Span Crossbeam Vibration Frequencies Analysis Based on an Analogous Beam Method

The novel method of an analogous beam is studied, which the flexural rigidity and mass per unit length correspond was described as the reciprocal of the mass per unit and the reciprocal of the flexural rigidity of the beam. It is shown that both beams possess the same natural frequencies of flexural vibration. In order to approximate calculation of these frequencies, the continuously distributed mass of the original beam is substituted for a number of concentrated masses. The analogous beam then becomes a chain of rigid links connected by pins and equipped with springs restraining the relative rotation of adjacent links. The equations of motion for the analogous beam can be solved by a procedure which consists of assuming a value for the natural frequency and calculating the deflections successively from one end of the beam to the other. Under normal circumstances, there will be a certain error, and one boundary condition will not be satisfied. The procedure is repeated with different values of the frequency until the error is removed. The method is illustrated by an example of a Crossbeam for which the fundamental frequency is found.

DSC analysis of a simply supported anisotropic rectangular plate

The discrete singular convolution (DSC) algorithm is used to analyze the deflection and free vibration behavior of a simply supported anisotropic rectangular plate. A novel approach is proposed to solve the difficulty in using DSC to handle the simply supported boundary conditions with bending–twisting coupling. DSC results are presented for bending under distributed load and a center concentrated load, and natural frequencies of flexural vibrations. It is shown that the DSC with proposed method to apply the simply supported boundary conditions yields very accurate results as compared to exact solutions or results obtained by methods of differential quadrature and finite element with fine meshes. It is also verified that neglecting the bending–twisting coupling in applying the simply supported boundary conditions may result incorrect solutions, especially for the bending analysis of anisotropic plates.

Fuzzy pattern recognition of impact acoustic signals for nondestructive evaluation

In order to prevent the danger caused by falling tiles from high-rise buildings, a rapid and effective non-destructive testing and evaluation (NDT&E) technique has been developed to assess the tile–wall bonding quality. The proposed technique is based on sounds excited by controlled impacts, and can easily be integrated with climbing robots to automate the tile–wall inspection process without the need for human workers to work at life-threatening height. To facilitate the result evaluation and maintenance planning, the approximate size of defective area needs to be assessed and determined. Though it is well-known that the natural frequency of flexural vibration is related to the size of the debonded area, strong multiple-mode frequencies caused by complex shapes of the defects or impacting at geometric edge (not at the center) of the defects impose significant difficulties in extracting automatically the natural frequency from the sound signals. In this paper, a fuzzy scheme is introduced to improve the robustness and accuracy of defects assessment. Based on fuzzy theory, vibration principle and human experience, a fuzzy logic model relating the characteristics of impact sounds to the approximate size of the defect is developed and utilized. The design of the fuzzy system, including membership functions and fuzzy reasoning rules, is also provided. To demonstrate the validity of the proposed method, experimental results obtained from physical tile walls are presented and discussed.

542 乗客による鉄道車両走行時の車体弾性振動減衰効果(自動車・鉄道車両の制震,OS-3 ダンピング,総合テーマ「伝統を,未来へ!」)

The effect of passengers boarding on a railway vehicle is usually considered as additional mass in designing carbody. This means that natural frequencies of flexural vibrations of a carbody are decreased as the number of passenger increases. However, previous researches, in which some stationary excitation tests were carried out for actual railway vehicles, indicate that on-board passengers behave not as additional mass but as damping. In this report, the authors examine the damping effect by passengers in running condition. A series of running test using a Shinkansen-train was conducted, and relatively large reduction of flexural vibration of carbody was observed when passengers were on-board. We also tried to imitate or simulate the damping effect by passengers using flexible polyethylene tanks filled with water. As a result of excitation test for a commuter-type test vehicle, at least two-different modes of flexural vibration were reduced when water tanks (total 630-1050 kg) were loaded.

Attenuation of flexural vibration for floating floor and floating box induced by ground vibration

This paper investigates the vibration isolation performance of floating floor and floating box structures to control rail vibration transmission. Simple theoretical and experimental methods are developed to analyze the effects of stiffener beam, mass and arrangement of isolator on the fundamental natural frequency of the flexural vibration of floating floor and box structure. The vibration reduction performances of floating floor and box structure are found to be degraded by flexural vibration of the floor or supporting stiffener beam. From the results of vibration measurements; stiffener beams increase the fundamental natural frequency of flexural vibration of floating floor and enhance vibration isolation. Also they can further alleviate the effect of flexural vibration using optimum isolator arrangement effectively. The proposed floating box design achieved a vibration reduction of 15–30 dB in frequency region of critical rail vibration (30–200 Hz).

Pultruded Glass Fiber-reinforced Plastic and Paperboard Composite Tubes

This research investigates the production of hybrid composite tubes by combining spirally wound paperboard tubes and fiber-reinforced polymer (FRP) materials using the pultrusion process to improve selected mechanical and physical properties of the paperboard tube. Higher natural frequencies of vibration of tubes used in the converting industry is needed in the marketplace because of continued pressure for throughput productivity improvements. This study focuses on hybrid tubes with increased natural frequencies of flexural vibration. A manufacturing feasibility study is conducted which determined that the pultrusion method is the most cost-effective process for this purpose. To evaluate the proposed method, a series of production trials is conducted using 1-in. (2.54 cm) diameter paperboard tubes as the internal mandrel with a thin layer (0.050 in. (0.13 cm)-0.15 in. (0.38 cm)) of FRP material pultruded on the outer surface. Trials are conducted with different FRP layer thicknesses, fiber architectures, and volume fractions. Bonding between the interface of the paperboard and the FRP layer is examined by a microscopy study, and flexural properties of the hybrid tubes are determined using modal analysis. Overall, the research project reveals that manufacturing composite FRP/paperboard tubes is feasible and that little modification is needed to the existing production machinery. In addition, the process results in a highly cost-effective method to produce hybrid paperboard and FRP tubes with high natural frequencies of vibration.

Combined numerical–experimental model for the identification of mechanical properties of laminated structures

A combined numerical–experimental method for the identification of six elastic material modulus of generally thick composite plates is proposed in this paper. This technique can be used in composite plates made of different materials and with general stacking sequences. It makes use of experimental plate response data, corresponding numerical predictions and optimisation techniques. The plate response is a set of natural frequencies of flexural vibration. The numerical model is based on the finite element method using a higher-order displacement field. The model is applied to the identification of the elastic modulus of the plate specimen through optimisation techniques, using analytical sensitivities. The validity, efficiency and potentiality of the proposed technique is discussed through test cases.

Vibration of Laminated Plates Having Elastic Edge Flexibilities

The first known investigation on the problem of free vibration of symmetric cross-ply laminated plates having elastically restrained edges is considered. The Ritz method is used in combination with a variational formulation and a first-order transverse shear deformation theory. The Ritz functions consist of polynomials that include a basic function that imposes the free edge boundary conditions. The elastic edge flexibilities are considered by both the linear elastic rotational and translational supports to act simultaneously. Numerical results are obtained to investigate the effects of elastic restraint stiffness, material properties, and geometric parameters upon the natural frequencies of flexural vibration.

VIBRATION ANALYSIS OF RECTANGULAR MINDLIN PLATES RESTING ON ELASTIC EDGE SUPPORTS

For the first time to the authors' knowledge, the problem of free vibration of a moderately thick rectangular plate with edges elastically restrained against transverse and rotational displacements is considered. The Ritz method combined with a variational formulation and Mindlin plate theory is used. The admissible functions consist of polynomials and basic functions that impose the required boundary conditions on the Mindlin plate. The applicability of the formulation is illustrated using three examples of plates with different combinations of elastically restrained edges and classical boundary conditions. Numerical results are obtained to investigate the effects of elastic spring stiffness, relative thickness and aspect ratio upon the natural frequencies of flexural vibration of rectangular Mindlin plates.

Modal Analysis of a Vibrating Plate Considering Internal Damping and Application to the Estimation of Radiated Sound Power. 1st Report, Derivation of the Relationship between Total Loss Factor and Frequency at the Natural Modes of Flexural Vibration.

The authors have proposed a basic theory for precisely estimating the frequency response of sound power radiating from plates due to their flexural vibration. In this theory, the equivalent viscous damping ratios of all the natural modes of flexural vibration are required to calculate total loss factor, radiation loss factor and driving point mobility, which are the three most important parameters for the estimation. At present the equivalent damping ratios are only determined using the experimental modal analysis of plates. However, if the damping ratios can be estimated by means of a theoretical method, the radiated sound power may be completely estimated by calculation. As the basis for that, the internal damping of metal is assumed to be viscous damping and is incorporated into the equation of flexural vibration of plates ; then, the relationship between the internal damping and the natural frequency of flexural vibration is derived using Mindlin's thick plate theory.

Dynamic Behavior of Thin-rimmed Helical Gears with Various Web Arrangements

This paper presents a study on dynamic behavior of thin-rimmed helical gears with various web arrangements. Natural frequencies of flexural vibration of gear body were measured and compared with the calculated results by Mindlin's method. The relations between these natural frequencies and spectra of circumferential, radial and axial vibrations were investigated. The circumferential, radial and axial vibration accelerations and the root stresses were measured under different running conditions by using a gear testing machine of a power absorbing type. On the basis of these results the effects of web arrangements on vibration and dynamic load were clarified to a considerable extent.

Rayleigh-Ritz vibration analysis of Mindlin plates

The Rayleigh-Ritz method is applied to the prediction of the natural frequencies of flexural vibration of square plates having general boundary conditions. The analysis is based on the use of Mindlin plate theory so that the effects of shear deformation and rotary inertia are included. The spatial variations of the plate deflection and the two rotations over the plate middle surface are assumed to be series of products of appropriate Timoshenko beam functions. Results are presented for a number of types of plate and these demonstrate the manner of convergence of the method as the number of terms in the assumed series increases.

The lateral vibration of slightly bent slender beams subject to prescribed axial end displacement

A simple analysis is presented for the prediction of the induced axial loads in initially slightly bent slender beams subject to prescribed axial end displacement. The effect upon the natural frequencies of flexural vibration of the beams is then determined. A simply supported beam and a fully clamped beam are considered and it is assumed that the initial deflection is of the same form as the first critical buckling mode for the straight beam. It is shown that the effect of relatively slight initial lack of straightness upon the induced axial load and bending natural frequencies is very significant. A practical application of the analysis would be in the prediction of the natural frequencies of vibration of heat exchanger tubing where the prescribed axial displacement takes the form of restrained thermal expansion and the results of a simple experiment concerning such are included.