Free 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.

Large Amplitude Free Flexural Vibrations of Functionally Graded Plates

The nonlinear formulation developed based on von Karman’s assumptions is employed to study the large free flexural vibration characteristics of functionally graded material (FGM) plates subjected to thermal environment. Temperature distribution is uniform over the plate surface and varied in thickness direction only. Material properties are assumed to be temperature dependent and graded in the thickness direction according to simple power law distribution. The nonlinear governing equations obtained using Lagrange’s equations of motion are solved using finite element procedure coupled with the direct iteration technique. The variation of nonlinear frequency ratio with amplitude is highlighted considering various parameters such as gradient index, temperature, thickness and aspect ratios, skew angle and boundary condition. For the numerical illustrations, silicon nitride/stainless steel is considered as functionally graded material. The results obtained here reveal that the temperature field and gradient index have significant effect on the large amplitude free flexural vibration of the functionally graded plate.

On the Vibrations and Thermal Buckling of Functionally Graded Spherical Caps

Here, the axisymmetric free flexural vibrations and thermal buckling characteristics of functionally graded spherical caps are investigated employing a three-noded axisymmetric curved shell element based on field consistency approach. The formulation is based on first-order shear deformation theory and it includes the in-plane and rotary inertia effects. The material properties are graded in the thickness direction according to the power-law distribution in terms of volume fractions of the constituents of the material. The effective material properties are evaluated using homogenization method. A detailed numerical study is carried out to bring out the effects of shell geometries, power law index of functional graded material and base radius-to-thickness on the vibrations and buckling characteristics of spherical shells.

Free flexural vibration of cracked composite laminated plate using higher-order XFEM

In this work, a higher-order XFEM approach is presented and implemented to analyze free flexural vibration in cracked composite laminated plate using higher-order shear deformation theory. The composite laminated plate experiences flexural vibration during its service life due to transverse shear loading. Out-of-plane modulus of the plate is much lesser than the in-plane modulus, so the free vibration study of the plate has become a crucial parameter for a safe service environment. The proposed methodology introduce higher-order tip enrichment functions (r2 terms) for crack simulation in laminated plate. These enrichment functions efficiently capture all conceivable displacements and rotations near the crack tip region. A comparative analysis between higher-order XFEM and conventional XFEM approach is presented in terms of computational accuracy. Several numerical examples are presented to demonstrate the accuracy of these higher-order enrichment functions, which contain variations in natural frequency with respect to crack sizes for different symmetric ply composite laminate plates under varied boundary conditions. The numerical results demonstrate that the proposed higher-order XFEM approach is significantly enhancing computational accuracy than the classical XFEM approach.

Damage Detection in Free–Free Glass Fiber Fabric Composite Beams by measuring Flexural and Longitudinal Vibrations

This paper presents the experimental investigation results of the use of the vibration method by impulse excitation of free flexural and longitudinal vibrations of composite materials. The purpose of the study is to establish the sensitivity of the method used for defect detection and localization. To realize the objective, rectangular notch-type defects were simulated at different distances and depths. The influence of the location and depth of the artificial cracks on the dynamic properties of the beams was investigated by measuring the natural frequencies of flexural and longitudinal vibrations. The conducted experiments show a change in the dynamic characteristics of the beam depending on the dimensions and location of the defects.

On the stochastic vibration analysis of the geometrically nonlinear graded cellular curved panels with material stochasticity

In this paper, an effort was made to study the free flexural vibration analysis of the geometrically nonlinear graded cellular curved panels in a stochastic domain. The nonlinear finite element formulations were developed using von Karman nonlinearity hypotheses based on higher-order shear deformation theory, including Lame's parameter. The first-order perturbation technique (FOPT) was employed to assess the stochastic vibration behavior of the graded cellular panels considering randomness in the material properties. The graded cellular curved panels are considered to have three distinguished porosity distributions. The properties of the graded cellular curved panels are continuously varying in the thickness direction as a function of porosity coefficient and mass density. The influences of material stochasticity, porosity coefficient, porosity distribution type, and geometric parameters have been included in the formulations. The detailed convergence and comparison studies were conducted to demonstrate the accuracy and efficiency of the present formulation and highlight the importance of the present study in stochastic environments. The various numerical examples for the second-order statistics (i.e., mean and standard deviation) of the linear frequency parameters (λ) and nonlinear frequencies of the graded cellular structures have been presented which may be considered as the guidelines for future studies in this area of research.

Free and forced small flexural vibrations of slightly curved slender composite beams with interlayer slip

This paper presents a beam theory for analyzing the dynamic bending response of slender slightly curved composite beams whose layers are flexibly connected and therefore subject to interlayer slip. The equations of motion and boundary conditions are derived using Hamilton’s principle, assuming separately for each layer the applicability of Euler–Bernoulli theory and a linear elastic relationship between the interlayer slip and the shear traction. For the problem of a three-layer slightly curved single-span beam with symmetric layer arrangement and soft-hinged bearings, analytical expressions for the natural frequencies and the eigenfunctions are derived. For the arbitrarily supported two-layer beam, on the other hand, a numerical solution scheme of the combined initial boundary value problem is presented. Several examples show how important it is to consider even very small deviations from the straight beam axis in the prediction of the dynamic response for slender beams with interlayer slip, in particular when all supports are immovable. The comparison of the beam solutions with the results of much more expensive FE analyses based on plane stress elasticity proves the accuracy of the presented theory.

Variable stiffness composite laminated beams - nonlinear free flexural vibration behavior using a sinusoidal based shear flexible structural theory accounting for Poisson’s effect

In the present work, formulation for the nonlinear free vibrational behavior of laminated variable stiffness composite beams with curvilinear fibers is made incorporating sinusoidal function for representing the transverse shear flexibility, geometrical nonlinearity, and accounting for Poisson’s effect through the constitutive equation for analyzing beams with general composite lay-up. Applying Hamilton’s principle combining with a finite element approach based on three-noded C1 beam element, the governing equations are formed through matrices. The solutions for the developed governing equations are evaluated iteratively by introducing eigenvalue analysis and the results are viewed through the frequency–amplitude relationship. A large number of design parameters like curvilinear fiber angles in a layer, number of layers, lay-up sequence, thickness ratio, etc. is assumed to visualize the nonlinear free vibration characteristics of VSCL beams. Also, for the practical situation, the influence of thermal environment and restraint elastically against beam ends rotation, to accommodate other than classical boundary conditions, hybrid beam (consisting of constant and variable stiffness layers) on the non-linear free vibrational behavior is presented.

Analytical investigation of the main cable bending stiffness effect on free flexural vibration of suspension bridges with a 3D cable system

This paper proposed a dimensionless continuum model of suspension bridges to comprehensively study the bending stiffness effect of the 3D cable system on the structural free flexural vibration, which has seldom been investigated in previous studies. In this model, coupled differential equations including the cable bending resistance term are derived via the Galerkin method. The equations are then nondimensionalized from a set of key design parameters, including the lateral inclination of hangers. The precision of the model is validated with the finite element (FE) method through an example. Then, parametric comparative studies of the effect under dimensionless parameters are conducted between 3D and 2D cable system suspension bridges. The results show that, the bending stiffness effects of the 3D cable system are more evident on the high order modes and antisymmetric modes of suspension bridges, which is similar with those effects of the 2D cable system. The hanger lateral inclination restrains the effect on the structural frequencies following a relationship of linear negative correlation; when the inclination angle varies from 30 to 60 degrees, the effect ratio of 3D to 2D cable system correspondingly varies from 0.33 to 0.67. Under a given hanger lateral inclination (48.6 degrees) with other identical conditions, the variation trend of the effects on the high order frequencies are similar between 3D and 2D cable systems; the effect value of the 3D cable system is on average 0.625 times that value of the 2D cable system.

Calibration of the length scale parameter for the stress-driven nonlocal elasticity model from quasi-static and dynamic experiments

The available experimental results in the literature on the quasi-static bending and free flexural vibration of microcantilevers and nanocantilevers are used to calibrate the length scale parameter of the stress-driven nonlocal elasticity model. The Bernoulli–Euler theory is used to define the kinematic field. The closed form solution derived for the bending problem is used to calibrate the length scale parameter by fitting the load–displacement curves to the experimental results. For the vibration problem, the calibration is done using the least-squares curve fitting method for the natural frequencies. The stress-driven nonlocal theory can adequately capture the size-dependent experimental results.

Free flexural vibration analysis of composite sandwich plate with reentrant honeycomb cores using homogenized plate model

The composite sandwich plate with reentrant honeycomb core (CSP-RHC) is a special auxetic composite structure, and the effect of a negative Poisson’s ratio (NPR) on its dynamic characteristics is unclear. In this article, the flexural vibration of CSP-RHC is analyzed based on the 2D homogenized plate model (2D-HPM) using the variational asymptotic method, and the effective plate properties are obtained by homogenizing the unit cell of CSP-RHC. The efficiency and precision of the free flexural vibration results using 2D-HPM are verified by comparison with 3D-FEM results under different boundary conditions, and the influences of the selective parameters on the free vibration characteristics of CSP-RHC are discussed. The results show that the natural frequencies of CSP-RHC decrease with the increase of the aspect ratio, but increase with the increase of the plate thickness and core thickness ratio. The natural frequency of the plate generally decreased with the increased cell angle, regardless of a positive or negative Poisson’s ratio. Furthermore, the natural frequencies of CSP-RHC are obviously lower than that of the traditional honeycomb sandwich plate due to the NPR effect.

Influence of microstructural defects on free flexural vibration of cracked functionally graded plates in thermal medium using XFEM

This paper presents the influence of microstructural defects on the vibration characteristics of the cracked functionally graded plate in a thermal environment. Extended finite element formulation is implemented to model the cracked functionally graded plate based on higher-order shear deformation theory (HSDT). The level set function is implemented to track the crack in the functionally graded plate domain. The crack in the functionally graded plate domain is addressed by enriching the primary variable with additional functions using the partition of unity technique. The power-law distribution is implemented for the gradation of the material properties along the thickness direction to form functionally graded material (FGM). The microstructural defects are assumed as porosities in the plate. The porosity index determines the porosity in the FGM plate. The convergence and comparative study have been performed to verify the current formulation’s efficacy, accuracy, and reliability. Further, various numerical examples are presented. The effects of different influential parameters like porosity, volume fraction index, crack length, thickness ratio, and boundary condition on the natural frequencies have been discussed in detail. The presented computational approach is accurate, efficient, and robust enough to investigate the vibration response of the cracked FGM porous plates, which can provide a reference for future research by researchers.

Free flexural vibrations of homogeneous beams with symmetrically variable depths

The subject of the paper are homogeneous beams of symmetrically variable depth and bisymmetrical cross sections. Free flexural vibrations of these beams are analytically and numerically studied. Based on Hamilton’s principle, the differential equations of motion of these beams are obtained. The equations of motion are analytically solved with consideration of the bending lines of these beams subjected to their own weight. The fundamental natural frequency for exemplary beams is derived and presented in Tables and Figures.

Three-dimensional free flexural vibrations of fluid-filled functionally graded circular cylindrical shell with curvilinear radius variation

The three-dimensional free flexural vibrations of functionally graded circular cylindrical shell with curvilinear radius variation fully or partially filled with incompressible and inviscid fluid are investigated for the first time using a coupled axisymmetric solid-fluid hierarchical finite element method based on curved elements. Curved edges are described exactly by means of blending functions. The method is used in conjunction with the three-dimensional elasticity theory and Laplace’s equation to obtain the governing equations. The material properties follow the power law distribution in the thickness direction. The effectiveness of the method is illustrated using examples of clamped-free and clamped-guided functionally graded circular cylindrical shells with quadratic and cubic radius variations. The method is validated through convergence test and comparison study. New results for the cycle frequencies of the lowest two flexural modes are presented, which may serve as benchmarks for future studies. Moreover, the effects of several geometrical and mechanical parameters on the cycle frequencies, modal deflections, and modal hydrodynamic pressures are explored.

Large Amplitude Free Vibration Analysis of Isotropic Curved Nano/Microbeams Using a Nonlocal Sinusoidal Shear Deformation Theory-Based Finite Element Method

In this investigation, the nonlinear flexural free vibration characteristics of size dependent curved isotropic nano/microbeams are studied using the nonlocal elasticity theory along with the sinusoidal shear deformation theory (SSDT). Based on Hamilton’s principle, the governing equations are derived in terms of generalized displacements using finite element approach. The formulation by extending the von-Karman model accounts for both large deflections and rotations in the strain displacement definition. Direct iterative procedure is used to solve the nonlinear problem through eigenvalue procedure. This investigation aims to study the effect of various design parameters like slenderness ratio of the beam, curved beam included angle, nonlocal parameter, and boundary conditions on the amplitude–frequency response obtained from the free vibration behavior of curved nano/microbeams. This study is very useful to designers for optimal design of nano/microsystem involving structural elements of the beam type.

Nonlinear flexural free vibrations of size-dependent graphene platelets reinforced curved nano/micro beams by finite element approach coupled with trigonometric shear flexible theory

In this work, the nonlinear flexural free vibration behavior of size-dependent curved nano/micro beams with reinforcement of graphene platelets is studied using the nonlocal elasticity theory along with a trigonometric shear flexible beam theory. The governing equations through the finite element model are derived in terms of displacements using the dynamic version of principle of virtual work. The formulation extends the von-Karman model to account for both large deflections and rotations in the strain displacement relationship. Direct iterative procedure is used to solve the nonlinear eigenvalue problem. This methodological study aims to analyze the influence of various design parameters like slenderness ratio of the beam, curved beam included angle, distribution pattern of porosity and graphene platelets, nonlocal parameter, and boundary conditions on the free vibration behavior of curved nano/micro beams by studying the amplitude frequency response curves. The present work also highlights the variation in modeshapes with reference to axial/transverse directions.

Free flexural vibration of thin-walled honeycomb sandwich cylindrical shells

The free flexural vibration of thin-walled honeycomb sandwich cylindrical shell was investigated by experiments and Flügge's shell theory simulations. The natural frequencies were measured and compared with simulations three-dimensional model. In the simulation analysis, the porous honeycomb core is simplified as a homogeneous orthotropic material, and its equivalent material properties are determined by the corrected Gibson's formula. Flügge's shell theory is compared with experimental results and finite element method calculation results, and the maximum relative error is only 4.749%.The comparison results show that Flügge's shell theory can accurately calculate the natural frequency of the thin honeycomb sandwich cylindrical shell.

Effects of the Variational Consistency and Variational Inconsistency on Free Flexural Vibration Frequencies of Simply-supported Rectangular Isotropic Shear Deformable Beams

In order to derive governing differential equations of a displacement-based shear deformation beam theory, one can either utilise the variationally consistent approach (VCA) or the variationally inconsistent approach (VIA). In the VCA, variational principles are utilised to derive beam governing equations. Whereas in the VIA, beam gross / integrated two-dimensional equilibrium equations are utilised to derive beam governing equations. Shimpi et al. [2] have proposed the single variable shear deformation beam theory (SVSDBT) which is based on the VIA. Whereas, Shimpi et al. [1] have proposed the two variable shear deformation beam theory (TVSDBT) which is based on the VCA. In the SVSDBT and TVSDBT, the beam axial displacement and beam transverse displacement consist of bending and shearing components. In both theories, bending components do not take part in the cross-sectional shearing force, and shearing components do not take part in the cross-sectional bending moment. Both theories utilise linear strain-displacement relations. Displacement functions of both theories give rise to the beam transverse shear strain (and hence to the beam transverse shear stress) which varies quadratically through the beam thickness and maintains transverse shear stress-free beam surface conditions. For the case of dynamics, the SVSDBT has only one governing equation, and the TVSDBT has two governing equations which are inertially coupled. The aim of this paper is to present the comparison study on free flexural vibration frequencies of simply-supported rectangular isotropic shear deformable beams obtained by utilising the SVSDBT and TVSDBT. These frequencies are obtained for various values of the beam thickness-to-length ratio and beam vibration mode parameter. This comparison shows that for both theories, the VCA and VIA have a minimal effect on resulting beam free flexural vibration frequencies.

Hybrid Sandwich Panels: A Review

A high specific stiffness, high specific strength, and tailoring the properties for specific application have attracted the attention of the researchers to work in the field of laminated composites and Sandwich structures. Rapid use of these laminated composites and Sandwich structures necessitated the development of new theories that suitable for the bending, buckling and vibration analysis. Many articles were published on free vibration of beams, plates, shells laminated composites and sandwich structures. In this article, a review on free vibration analysis of shear deformable isotropic beams, plates, shells, laminated composites and sandwich structures based on various theories and the exact solution is presented. In addition to this, the literature on finite element modeling of beams, plates, shells laminated composites and sandwich structures based on classical and refined theories is also reviewed. The present article is an attempt to review the available literature, made in the past few decades on free flexural vibration response of Fiber Metal laminated Composites and Sandwich panels using different analytical models, numerical techniques, and experimental methods.

Damping analysis of nonwoven natural fibre-reinforced polypropylene composites used in automotive interior parts

The aim of this work is to evaluate the dynamic properties of nonwoven flax, hemp, kenaf and glass fibre-reinforced polypropylene (PP) composites. Also, the influence of some parameters, such as the type of reinforcement, the fibre/matrix weight ratio, the fibre orientation and the porosity content, on the damping behaviour of these nonwoven composites is investigated. To this end, a free flexural vibrations analysis was conducted to experimentally identify their natural frequencies and their associated loss factors. The obtained results show that the nonwoven composites reinforced by natural fibres present higher loss factors compared with those of the glass-PP composite. These interesting damping properties make these nonwoven composites very attractive for automotive applications where the dissipation of vibrations is highly requested.