Free Transverse Vibration Research Articles

Estimation of damping in riveted short cantilever beams

In layered and riveted structures, vibration damping happens because of a micro slip that occurs because of a relative motion at the common interfaces of the respective jointed layers. Other parameters that influence the damping mechanism in layered and riveted beams are the amplitude of initial excitation, overall length of the beam, rivet diameter, overall beam thickness, and many layers. In this investigation, using the analytical models such as the Euler–Bernoulli beam theory and Timoshenko beam theory and half-power bandwidth method, the free transverse vibration analysis of layered and riveted short cantilever beams is carried out for observing the damping mechanism by estimating the damping ratio, and the obtained results from the Euler–Bernoulli beam theory and Timoshenko beam theory analytical models are validated by the half-power bandwidth method. Although the Euler–Bernoulli beam model overestimates the damping ratio value by a very less fraction, both the models can be used to evaluate damping for short riveted cantilever beams along with the half-power bandwidth method.

Free Vibration Analysis of Axially Functionally Graded (AFG) Cantilever Columns of a Regular Polygon Cross-Section with Constant Volume

This paper deals with the transverse free vibration of axially functionally graded (AFG) cantilever columns under the influence of axial compressive load. The columns possessing a regular polygon in their cross-section are tapered and their material properties vary along the axis of the column. An emphasis is placed on the columns with constant volume for admissible geometries and materials. The governing differential equation of the problem is derived and solved using the direct integral approach in conjunction with the determinant search technique. The obtained results are in good agreement with those in the available literature and computed by finite element analysis. Numerical examples for the natural frequency and mode shape of the columns are presented to investigate the effects of parameters related to geometrical nonuniformity and material inhomogeneity.

Free vibration and wave propagation of axially moving functionally graded Timoshenko microbeams

The transverse free vibration and wave propagation of functionally graded microbeams with an axial motion are investigated based on a nonlocal theory and the Timoshenko beam model. It is assumed the material properties of functionally graded microbeams vary along the thickness direction. The neutral plane of functionally graded materials is introduced, and the inhomogeneity of functionally graded Timoshenko microbeams is considered. The governing equations are derived using Hamilton’s principle, and the differential quadrature method is utilized to determine the first three-order natural frequencies of the microbeams with simply supported and clamped boundary conditions, respectively. The effects of gradient index, nonlocal parameter and axial velocity on natural frequencies are investigated. Moreover, the wave propagation characteristics of functionally graded Timoshenko microbeams are analyzed, and the significant influences of wave number and other variables on wave propagation frequencies and wave velocities are studied. The different influence patterns of nonlocal effect are observed in transverse vibration and wave propagation. The nonlocality shows a weakening phenomenon in transverse vibration of axially moving functionally graded Timoshenko microbeams, while it reveals both weakening and strengthening phenomena in wave propagation. Therefore, two kinds of existing nonlocal scale effects are further confirmed and this is an additional contribution of the present paper.

Bilaterally flexural vibrations and instabilities of moving piezoelectric nanowires with surface effect

Vibrations of piezoelectric beam-like nanostructures have been widely examined; however, their dynamic response and instabilities in the moving state accounting for surface effect have not been investigated yet thoroughly. Given the importance of the subject and potential applications of piezoelectric nanowires in moving bodies, herein, we are interested in exploring their bidirectional-transverse vibrations in the context of the surface elasticity theory of Gurtin and Murdoch. To this end, equations of motion that describe transverse vibrations of axially moving piezoelectric nanowires are obtained by employing Rayleigh’s and Timoshenko’s beam theories. Using assumed mode approach in conjunction with Galerkin method, the flexural frequencies are evaluated and the explicit expressions of divergence velocities accounting for surface energy are derived. The influences of nanowire’s length, axial velocity, surface energy, shear deformation effect, and voltage difference on the free transverse vibrations as well as divergence and flutter instabilities of the moving piezoelectric nanostructure are displayed. The present work can provide a solid basis for better understanding of vibrations of moving vertically aligned ensembles of piezoelectric nanowires. An axially moving piezoelectric nanowire.

Transverse free vibration of Euler-Bernoulli beam with pre-axial pressure resting on a variable Pasternak elastic foundation under arbitrary boundary conditions

Transverse free vibration of Euler-Bernoulli beam with pre-axial pressure resting on a variable Pasternak elastic foundation under arbitrary boundary conditions

Study on Natural Frequencies of Transverse Free Vibration of Functionally Graded Axis Beams by the Differential Quadrature Method

Functionally gradient materials with special mechanical characteristics are more and more widely used in engineering. The functionally graded beam is one of the commonly used components to bear forces in the structure. Accurate analysis of the dynamic characteristics of the axially functionally graded (AFG) beam plays a vital role in the design and safe operation of the whole structure. Based on the Euler-Bernoulli beam theory (EBT), the characteristic equation of transverse free vibration for the AFG Euler-Bernoulli beam with variable cross-section is obtained in the present work, and the governing equations of the beam are transformed into ordinary differential equations with variable coefficients. Using differential quadrature method (DQM), the solution formulas of characteristic equations under different boundary conditions are derived, and the natural frequencies of the AFG beam are calculated, while the node partition of a non-uniform geometric progression is discussed.

Small scale and spin effects on free transverse vibration of size-dependent nano-scale beams

This study aims to investigate the transverse vibration of size-dependent nano-beam which spins about its longitudinal axis. To this end, nano-beams are modeled based on the Euler–Bernoulli and Timoshenko theories. To consider the size dependency of the nano-beams, the nonlocal elasticity theory is used. By using an exact analytical solution, the governing equations of motions are solved. Comprehensive results are presented to show the effects of various boundary conditions, nonlocal scale parameter, rotational speed, and geometrical parameters on natural frequencies and critical speeds of spinning nano-beam. As spinning nano-beams are the main part of any rotary nano-machines, conclusions of present analysis promote researcher’s information about the vibrational behavior of spinning nano-beams.

DQEM analysis of free transverse vibration of rotating non-uniform nanobeams in the presence of cracks based on the nonlocal Timoshenko beam theory

This paper deals with the free transverse vibration characteristics of a rotating non-uniform nanocantilever with multiple open cracks. Employing Eringen’s nonlocal elasticity and the Timoshenko beam theory, the non-dimensional governing differential equations for the above-mentioned problem are derived. The cracked beam is divided into intact sub-beams between two subsequent cracks connected by linear and rotational springs. Differential quadrature element method is utilized to solve the established governing equations of motion of each segment, along with the corresponding boundary conditions and compatibility conditions at the cracked sections. The frequency parameters and vibration modes of the rotating cracked beam for different crack positions and severities under various nonlocal, geometric and dynamic conditions are studied, and the relevant graphs are plotted. Since rotating nanocantilevers are found mostly as blades of rotating nanodevices, the results can provide useful guidance for the study and design of the next generations of nanoturbines, nanogears etc.

Comparison of machine learning methods for crack localization

In this paper, the Haar wavelet discrete transform, the artificial neural networks (ANNs), and the random forests (RFs) are applied to predict the location and severity of a crack in an Euler–Bernoulli cantilever subjected to the transverse free vibration. An extensive investigation into two data collection sets and machine learning methods showed that the depth of a crack is more difficult to predict than its location. The data set of eight natural frequency parameters produces more accurate predictions on the crack depth; meanwhile, the data set of eight Haar wavelet coefficients produces more precise predictions on the crack location. Furthermore, the analysis of the results showed that the ensemble of 50 ANN trained by Bayesian regularization and Levenberg–Marquardt algorithms slightly outperforms RF.

Transverse free vibration analysis of a tapered Timoshenko beam on visco-Pasternak foundations using the interpolating matrix method

The characteristics of transverse free vibration of a tapered Timoshenko beam under an axially conservative compression resting on visco-Pasternak foundations are investigated by the interpolating matrix method. The research is executed in view of a three-parameter foundation which includes the effects of the Winkler coefficient, Pasternak coefficient and damping coefficient of the elastic medium. The governing equations of free vibration of a non-prismatic Timoshenko beam under an axially conservative force resting on visco-Pasternak foundations are transformed into ordinary differential equations with variable coefficients in light of the bending rotation angle and transverse displacement. All the natural frequencies orders together with the corresponding mode shapes of the beam are calculated at the same time, and a good convergence and accuracy of the proposed method is verified through two numerical examples. The influences of foundation mechanical characteristics together with rotary inertia and shear deformation on natural frequencies of the beam with different taper ratios are analyzed. A comprehensive parametric numerical study is carried out emphasizing the primary parameters that describe the dynamic property of the beam.

Evaluation of modulus of elasticity and modulus of rupture of full-size wood composite panels supported on two nodal-lines using a vibration technique

The objective of this study was to determine the feasibility of using a transverse free vibration testing method to nondestructively evaluate the bending stiffness and strength of full-size (1220 × 2440 mm) wood composite panels (WCPs). Based on the principle of transverse free vibration of a thin plate, the dynamic modulus of elasticity (MOE) of a full-size WCP supported on two nodal-lines was derived using the Rayleigh method. Transverse free vibration test was conducted on 54 full-size medium density fiberboard (MDF) panels and 36 full-size particleboard (PB) panels under a two-nodal-line support. The modulus of elasticity (MOE) and modulus of rupture (MOR) of the panels were then determined by conducting static bending tests on the small samples cut from each panel. The results indicated that the transverse free vibration of a full-size panel supported on two nodal-lines was a valid testing approach for rapid determination of dynamic modulus of elasticity of full-size MDF and PB panels. Strong linear relationships were found between the dynamic MOE of full-size panels and static MOE of small samples, with the dynamic MOE approximately 8% higher than the static MOE. In addition, the two-variate models with dynamic MOE and thickness as predicting variables and the three-variate models with natural frequency, density and thickness as variables were found both effective in predicting the modulus of rupture of full-size MDF and PB panels.

Experimental study on the fundamental frequency of prestressed concrete bridge beams with parabolic unbonded tendons

In this study, a laboratory experiment–based testing program was conducted on a large-scale high-strength Prestressed Concrete I (PCI) beam with a parabolic unbonded tendon, capable of simulating a typical prestressed bridge member. Specifically, the simply supported PCI beam was subjected to free transverse vibrations with different prestress forces to demonstrate that its fundamental frequency was unaffected by such force. A reference model, describing the behavior of the PCI beam as a combination of two substructures interconnected, i.e., a compressed concrete Euler–Bernoulli beam and a tensioned parabolic cable, predicts no change in fundamental frequency with increasing prestress force when variation of the concrete's initial elastic modulus over time is taken into account. The large-scale experimental results confirmed that fundamental frequency is not an appropriate parameter for prestress loss prediction in concrete bridge beams with parabolic unbonded tendons. Accordingly, subsequent studies will be conducted for improving a static nondestructive testing method for such detection in concrete bridges.

Influence of beam butt face nonparallelism on free damped vibration spectrum of the beam

Introduction. The free vibration method is widely applied to detecting internal defects in rod and lamellar structural elements. In this work, it is used to diagnose a rod sample defect consisting in nonparallelism of the rod ends. The possibility of identifying a defect in the form of a bevel of one of the rod butts by the frequency spectrum of free damped longitudinal and transverse vibration after impact actions to the rod side or butt face is considered.
 Materials and methods. An experimental facility and a method of contact-free recording of the frequency spectrum of the rod natural vibration by its acoustic radiation spectrum are presented. Signs of a defect in the form of a split frequency of the rod transverse vibration are detected. At the same time, the maximum amplitudes of the split frequencies correspond to higher or lower frequencies depending on the face to which the impact is applied. This allows not only determining the presence of a defect in the form of the bevelled butt face, but also establishing its orientation relative to the side rod faces.
 Results. An approximate theoretical model is suggested. It explains the effect of frequency splitting in the presence of nonparallelism of the rod butts. For this, the frequency spectra of transverse vibration of two rods with lengths equal to the lengths of the smaller and the larger faces of the original rod with the bevel edge are considered. Experiments show that the differences in the natural frequencies of the bending vibration of these rods, which correspond to the same eigenmodes, are consistent with the differences in the split frequencies of the free damped transverse vibrations of the bevelled-face rod. At this, relationship between the split frequency amplitudes allows determining not only the presence of bevelled face, but also its location.
 Conclusions. The application and development of the method through the analysis of the natural vibration spectrum can lead to creation of remote quality control equipment for rod structural elements.

Approach to establish a hybrid intelligent model for crack diagnosis in a fix-hinge beam structure

Purpose With the development of research toward damage detection in structural elements, the use of artificial intelligent methods for crack detection plays a vital role in solving the crack-related problems. The purpose of this paper is to establish a methodology that can detect and analyze crack development in a beam structure subjected to transverse free vibration. Design/methodology/approach Hybrid intelligent systems have acquired their own distinction as a potential problem-solving methodology adopted by researchers and scientists. It can be applied in many areas like science, technology, business and commerce. There have been the efforts by researchers in the recent past to combine the individual artificial intelligent techniques in parallel to generate optimal solutions for the problems. So it is an innovative effort to develop a strong computationally intelligent hybrid system based on different combinations of available artificial intelligence (AI) techniques. Findings In the present research, an integration of different AI techniques has been tested for accuracy. Theoretical, numerical and experimental investigations have been carried out using a fix-hinge aluminum beam of specified dimension in the presence and absence of cracks. The paper also gives an insight into the comparison of relative crack locations and crack depths obtained from numerical and experimental results with that of the results of the hybrid intelligent model and found to be in good agreement. Originality/value The paper covers the work to verify the accuracy of hybrid controllers in a fix-hinge beam which is very rare to find in the available literature. To overcome the limitations of standalone AI techniques, a hybrid methodology has been adopted. The output results for crack location and crack depth have been compared with experimental results, and the deviation of results is found to be within the satisfactory limit.

Multiparametric Analytical Solution for the Eigenvalue Problem of FGM Porous Circular Plates

Free vibration analysis of the porous functionally graded circular plates has been presented on the basis of classical plate theory. The three defined coupled equations of motion of the porous functionally graded circular/annular plate were decoupled to one differential equation of free transverse vibrations of plate. The one universal general solution was obtained as a linear combination of the multiparametric special functions for the functionally graded circular and annular plates with even and uneven porosity distributions. The multiparametric frequency equations of functionally graded porous circular plate with diverse boundary conditions were obtained in the exact closed-form. The influences of the even and uneven distributions of porosity, power-law index, diverse boundary conditions and the neglected effect of the coupling in-plane and transverse displacements on the dimensionless frequencies of the circular plate were comprehensively studied for the first time. The formulated boundary value problem, the exact method of solution and the numerical results for the perfect and imperfect functionally graded circular plates have not yet been reported.

Natural frequency of axially functionally graded, tapered cantilever beams with tip masses

This article presents a general solution for the free transverse vibration of non-uniform, axially functionally graded cantilevers loaded at the tips with point masses. Attention is focused on undamped cantilevers that fall within the range of validity of Euler-Bernoulli beam theory. The Myklestad Method is shown to be an efficient and accurate tool for this purpose. Results are presented for a number of cases of different beam geometries and material gradients.

A unified procedure for free transverse vibration of rectangular and annular sectorial plates

A unified solution procedure applicable for analyzing the free transverse vibration of both rectangular and annular sectorial plates is presented in this study. For the annular sectorial plate, the basic theory is simplified by a variable transformation in the radial direction. The analogies of coordinate system, geometry and potential energy between the two different shapes are drawn and then unified in one framework by introducing the shape parameter. A generalized solving procedure for the two shapes becomes feasible under the unified framework. The solution adopts the spectro-geometric form that has the advantage of describing the geometry of structure by mathematical or design parameters. The assumed displacement field and its derivatives are continuous and smooth in the entire domain, thereby accelerating the convergence. In this study, the admissible functions are formulated in simple trigonometric forms of the mass and stiffness matrices for both rectangular and annular sectorial plates can be obtained, thereby making the method computationally effective, especially for analyzing annular sectorial plates. The generality, accuracy and efficiency of the unified approach for both shapes are fully demonstrated and verified through benchmark examples involving classical and elastic boundary conditions.

APPLICATION OF FINITE ELEMENT METHOD IN SOLVING THE VIBRATION PROBLEM OF CIRCULAR PLATES

The finite element method is applied to the free transverse vibration of circular plate. The assumptions of the classical plate theory are app_lied in deriving the basic equations required for the solution of the problem. The triangular bending element used in describing the bending of the plate is characterized with only nine degrees of freedom. Consequ-ently the transverse deformation of the element is expressed by a poly-nomial interpolation function of third order. Using the area coordinate system, the expression for each of stiffness and mass matrices is derived in more convenient form. Simplification processes are used in the derivation of these matrices using a transfor-mation matrix. A complete modal analysis for the plate is obtained by a constructed computer program written with FORTRAN-IV language. Fortunately this triangular element gave good results with respect to those obtained by the exact method.

Exact mathematical solution for nonlinear free transverse vibration of beams

In the present paper, an exact mathematical solution has been obtained for nonlinear free transverse vibration of beams, for the first time. The nonlinear governing partial differential equation in un-deformed coordinates system has been converted in two coupled partial differential equations in deformed coordinates system. A mathematical explanation is obtained for nonlinear mode shapes as well as natural frequencies versus geometrical and material properties of beam. It is shown that as the th mode of transverse vibration excited, the mode 2 th of in-plane vibration will be developed. The result of present work is compared with those obtained from Galerkin method and the observed agreement confirms the exact mathematical solution. It is shown that the governing equation is linear in the time domain. As a parameter, the amplitude to length ratio has been proposed to show when the nonlinear terms become dominant in the behavior of structure.

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.