Bending Natural Frequencies Research Articles

Vibration Analysis of Functionally Graded Timoshenko Beams

The vibration behavior of a functionally graded Timoshenko beam is investigated by applying the transformed-section method. The material properties of a functionally graded (FG) beam are assumed to vary across the thickness according to a simple power law. The cross section of FG beam with two constituents is first transformed into an equivalent cross section of the material on the top. Then, the lateral and longitudinal vibration equations of a homogeneous Timoshenko beam are separately applied to the beam with the transformed section. The bending natural frequencies of FG beam are evaluated using the Chebyshev collocation method, and the longitudinal natural frequencies are also obtained from the known closed-form solutions. Some of the analytical results are compared with the existing numerical data to validate the present model accuracy. Good agreement has been observed between the analytical and numerical data. The effects of aspect ratio, volume fraction, and boundary conditions on the free-vibration behavior of FG beam are discussed. The present analytical solutions provide an insight to the effects of various parameters on the vibration behavior of the beam. They also serve as benchmarks for testing the vibration results obtained by other analytical or approximate methods.

A generally modified cuckoo optimization algorithm for crack detection in cantilever Euler-Bernoulli beams

In this research, a new and Generally Modified Cuckoo Optimization Algorithm (GMCOA) for solving the problem of detecting and estimating open-edge cracks in cantilever Euler-Bernoulli beams is proposed. This method is generated via applying general changes to Cuckoo Optimization Algorithm (COA) and improving its performance. The performance of this algorithm is investigated by performing simulation tests on a cantilever beam. Frequency response equations of a cracked beam is modelled using Euler-Bernoulli beam method and Hamilton's principle and its results are verified via modal tests on a sample beam. Tuning parameters of this optimization procedure is determined using Taguchi design of experiments method. An objective function consisting of the difference between measured and calculated first to fourth bending natural frequencies of the cracked beam is considered for optimization. Analyzing and comparing the results obtained using this method and the results of previous researches shows a better performance for this method. This method, in addition to estimating crack depth and location more accurately than other methods, achieves this accuracy with lower number of Function Evaluations (FEs). By using this number of FEs, other methods have not found crack location and depth more accurately than this method.

Bending and free vibration analysis of nanobeams by differential and integral forms of nonlocal strain gradient with Rayleigh–Ritz method

The static bending and free vibration behavior of Euler nanobeams are studied via nonlocal strain gradient elasticity. Three different approaches to nonlocal strain gradient are investigated including differential form of nonlocal strain gradient, integral nonlocal strain gradient with satisfying higher order boundary conditions and basic form of integral nonlocal strain gradient without satisfying higher order boundary conditions. Two different types of Rayleigh–Ritz method are adopted; in one of them, polynomial is chosen as admissible function and in the other, combination of polynomial and trigonometric are regarded to obviate ill-conditioning and achieve more accurate results in integral nonlocal elasticity in which high number of terms is needed to extract the results with desired accuracy. Next, the bending deflections and natural frequencies of nanobeams with different boundary conditions are obtained. The results show that using integral form of nonlocal strain gradient obviates the paradoxes of differential form. In addition, significant differences are observed between the results. Moreover, comparison between two different integral forms indicates that there are considerable discrepancies, especially in the cases of high material characteristic parameter with low nonlocal parameter. Advantage of using the basic form of nonlocal strain gradient is that it does not need to definition and satisfying the higher order boundary conditions.

The effect of prestress force magnitude on the natural bending frequencies of the eccentrically prestressed glass fibre reinforced polymer composite beams

This paper studies the effect of prestress force magnitude on natural frequencies and dynamic behaviour of eccentrically prestressed glass fibre reinforced polymer composite beams, including the theoretical background, numerical results and experimental verification. The term prestress indicates the initial tensile stress applied to the fibres embedded in selected external layers of the composite material. First, the paper presents the theoretical background of the finite element method modelling of prestressed composites. Then, the results of numerical simulations conducted for a five-layered glass-epoxy composite beam are presented. The natural frequencies corresponding to three initial bending modes are analyzed for different prestressing force levels and for different fibre volume content. Finally, the results are verificated by experimental modal analysis conducted on three different glass-epoxy composite specimens of various mechanical parameters. Both the numerical results obtained from finite element method and the experimental results obtained from experimental modal analysis reveal that the first bending frequency increases and the two subsequent bending frequencies decrease due to the prestressing force. The comparison of numerical and experimental data confirms the effect and allows to quantify the influence that the prestress force has on the natural frequencies of composites, which is an interesting and practically relevant phenomenon.

Estimation of efficiency of vibration damage detection in stepped shaft of steam turbine

The vibration based damage detection has practically no alternative as applied to the rotating shafts of steam turbines during operation. However, taking into account the heavy operating conditions of turbines and the complex shape of the turbines shaft, the choice of most effective method of vibration diagnostics is not an easy problem. To solve this problem, there was developed a relatively simple procedure of comparative evaluation of the effectiveness of vibration diagnostics of damage. The procedure is based on the determination of the change of shaft’s compliance caused by a crack with the use of linear fracture mechanics. It was demonstrated as applied to the stepped rotors and shafting of steam turbine. The comparative analysis of efficiency of vibration diagnostics was performed based on the change of natural bending, longitudinal or torsional frequency of vibration to detect transverse or longitudinal crack in rotors and shafting of steam turbine K-200-130.

Predictions of the frequencies of bending-torsion coupled laminated composite plates with discontinuities: Novel analytical modeling and experimental validation

A novel analytical model that accurately represents the free vibration response of intact/damaged composite plates accounting for the bending-torsional coupling is developed. Unlike the existing models for the bending-torsion vibrations of composite plates, in the present model, the equations of motion are derived in an uncoupled form. It is then easier to derive analytical solutions for composite plates under coupled bending-torsion vibrations. In addition, for the first time, analytical solutions for the free bending and torsional vibrations of cantilever and clamped-clamped laminated composite plates with discontinuities including circular holes, rectangular holes, notches, and internal and edge cracks are easily obtained. Shifts in the natural frequencies due to discontinuities are modeled where the reductions in the plate stiffnesses are related to the amount of the energy release rate due to a defect formation. In order to verify the accuracy of the proposed model and the derived solutions, the proposed model is compared to the available conventional models. Moreover, experimental measurements of the free bending and torsional vibrations of carbon-fiber-laminated composite plates with circular holes is performed. A set of finite element analyses are carried out and compared to the experimental and the proposed model results. Then, a parametric study is presented in order to investigate the effects of the discontinuity size and location on the bending and torsional natural frequencies of laminated composite plates with various forms of discontinuities. The performed analyses demonstrate the accuracy of the proposed models to represent the free vibrations of laminated composite plates with discontinuities.

Nonlinear mechanics of doubly curved shallow microshells

This paper performs a thorough investigation on the nonlinear size-dependent bending characteristics and natural frequencies of doubly curved shallow microshells. A nonlinear continuous model for a general doubly curved microshell is developed on the basis of Donnell's nonlinear shell theory and in the framework of the modified couple-stress strain gradient theory. In particular, the doubly curved microshell equations of motion of partial differential type are derived while accounting for geometric nonlinearities and small-scale effects. The continuous model is transformed into a discretised set of equations via application of the two-dimensional Galerkin technique. A large number of modes are retained in both linear and nonlinear investigations of microshells to ensure converged and reliable results. The linear natural frequencies are reported for various microshells of rectangular and square bases. The nonlinear static deflection curves for both in-plane and out-of-plane displacements are constructed and the effects of different parameters, such as the radius of curvature, sign of the radius of the curvature, and the length-scale parameter are examined.

Optimizing the static–dynamic performance of the body-in-white using a modified non-dominated sorting genetic algorithm coupled with grey relational analysis

ABSTRACTThis article presents a hybrid method combining a modified non-dominated sorting genetic algorithm (MNSGA-II) with grey relational analysis (GRA) to improve the static–dynamic performance of a body-in-white (BIW). First, an implicit parametric model of the BIW was built using SFE-CONCEPT software, and then the validity of the implicit parametric model was verified by physical testing. Eight shape design variables were defined for BIW beam structures based on the implicit parametric technology. Subsequently, MNSGA-II was used to determine the optimal combination of the design parameters that can improve the bending stiffness, torsion stiffness and low-order natural frequencies of the BIW without considerable increase in the mass. A set of non-dominated solutions was then obtained in the multi-objective optimization design. Finally, the grey entropy theory and GRA were applied to rank all non-dominated solutions from best to worst to determine the best trade-off solution. The comparison between the GRA and the technique for order of preference by similarity to ideal solution (TOPSIS) illustrated the reliability and rationality of GRA. Moreover, the effectiveness of the hybrid method was verified by the optimal results such that the bending stiffness, torsion stiffness, first order bending and first order torsion natural frequency were improved by 5.46%, 9.30%, 7.32% and 5.73%, respectively, with the mass of the BIW increasing by 1.30%.

Design of single‐span beams for SLS and ULS using semi‐continuous beam‐to‐column joints

AbstractThe objective of this article is to facilitate the use of semi‐continuous joints by providing ready‐to‐use tables and charts for design, which also permit the benefits accruing from the use of semi‐continuous composite beams to be directly identified. The relevant features for the design of composite beams, such as bending moment distribution, maximum deflection and natural frequency, are derived analytically. The influence of semi‐continuous steel joints on the performance of single‐span composite beams is analysed. The article concludes with an example of the use of a new slim‐floor beam type – the composite slim‐floor beam (CoSFB). The tables and charts derived are applied in this example with the intention of demonstrating the advantages of using semi‐continuous joints in combination with composite beams. The present article extends the design of semi‐continuous steel beams presented in the previous article on this subject [1].

The effect of carbon nanotube dispersion on the dynamic characteristics of unidirectional hybrid composites: An experimental approach

Abstract An experimental study on the natural frequencies and damping properties of hybrid composite beams and plates are presented in this paper. Hybrid composites consist of a matrix material along with both nano-fillers and micro fibers. In this research, carbon nanotube (CNT) is used as nano-filler. Epoxy, CNT/epoxy, glass/epoxy, CNT/glass/epoxy, carbon/epoxy and CNT/carbon/epoxy beam-shaped specimens are fabricated. Free vibration analysis of cantilever beams is examined in order to characterize the dynamic behavior of the specimens. 9.4% increase in the fundamental frequency and 12.3% decrease in the damping ratio of the CNT/glass/epoxy beam are observed with respect to those of the glass/epoxy beam specimen. Further, it is observed that CNT/carbon/epoxy beam has degraded fundamental frequency (13.9%) and enhanced damping ratio (31.5%) in comparison with those of carbon/epoxy beam specimen. Moreover, the first bending natural frequencies of the glass/epoxy, CNT/glass/epoxy, carbon/epoxy and CNT/carbon/epoxy plates are obtained experimentally. Plates with all free edges (FFFF) and plates with one edge clamped (CFFF) are investigated. Finite element models confirm the results obtained. Furthermore, the effects of the interfacial adhesion between the CNT/epoxy matrix and micro fibers on the damping properties of the laminated composites are discussed using the obtained results and the scanning electron microscopy (SEM) images.

Free vibration study of v-shape and rectangular shape double-sided cracks in a cantilever beam

The vibration response of a structural member changes due to the presence of crack because crack introduced local flexibility in it. Depending upon the vibration amplitude the crack may be of open or close type. Crack gives catastrophic failure to the structure hence it is very crucial to understand the dynamic of the structure. In the inverse problem, vibrating properties like natural frequency, resonance amplitude can be used as a basic criterion in crack detection or in design of structures. Previously, a mathematical model was developed by W. M. Ostachowicz and M. Krawczuk for the cantilever beam which has two open double-sided v-shape cracks. This model is used to find the natural frequency and characteristics roots in bending mode. The aim of this study is to prove whether the developed mathematical model can be used for rectangular shape cracks because in various applications existence of v shape and rectangular shape cracks are very general. For this reason, results of v-shape cracked cases are used as a reference model. Modal analysis is done by using ANSYS software to get the pure bending natural frequencies of the various cracked cases. Then the same mathematical model is used for rectangular shape cracked cases to calculate the characteristics roots. The value of characteristics roots obtained for a v-shape crack by W. M. Ostachowicz and M. Krawczuk and a rectangular shape crack cases studied are in good agreement. Hence it proves the flexibility of the model developed with v-shape cracks. In addition to this, the effect of two double-sided cracks on the characteristics roots is studied for all the cracked cases given in the reference model. It is found that as crack depth increases at any unique location then natural frequency decreases. At last location even though crack depth increases value of natural frequency almost remains constant.

Acoustic Emission and Modal Frequency Variation in Concrete Specimens under Four-Point Bending

The Acoustic Emission (AE) and Dynamic Identification (DI) techniques were applied simultaneously, in an original way, to examine the stress dependent damage progress in pre-notched concrete beams tested in four-point bending. The damage mechanisms were characterized by analyzing the AE signals registered during the tests, conducted by increasing the specimen’s vertical deflection. In particular, the dominant fracture mode was identified, and correlations between dissipated and emitted energies were investigated. Moreover, variations in the natural bending frequencies, produced by the crack advancement under loading, were detected and put in relation with the cumulated AE energy. Two different types of piezoelectric (PZT) sensors, operating in well distinct frequency ranges, were used to measure AE and modal signals. This study may be of interest with an outlook on possible correlations between a multi-parameter structural monitoring and the solution of inverse problems by numerical models.

Static and Dynamic Mechanical Properties of Long-Span Cable-Stayed Bridges Using CFRP Cables

The elastic modulus and deadweight of carbon fiber-reinforced polymer (CFRP) cables are different from those of steel cables. Thus, the static and dynamic behaviors of cable-stayed bridges using CFRP cables are different from those of cable-stayed bridges using steel cables. The static and dynamic performances of the two kinds of bridges with a span of 1000 m were studied using the numerical method. The effects of geometric nonlinear factors on static performance of the two kinds of cable-stayed bridges were analyzed. The live load effects and temperature effects of the two cable-stayed bridges were also analyzed. The influences of design parameters, including different structural systems, the numbers of auxiliary piers, and the space arrangement types of cable, on the dynamic performance of the cable-stayed bridge using CFRP cables were also studied. Results demonstrate that sag effect of the CFRP cable is much smaller than that of steel cable. The temperature effects of CFRP cable-stayed bridge are less than those of steel cable-stayed bridge. The vertical bending natural vibration frequency of the CFRP cable-stayed bridge is generally lower than that of steel cable-stayed bridge, whereas the torsional natural vibration frequency of the former is higher than that of the latter.

Dynamic Response Analysis of Simply Supported Steel Beam Under Natural Fire Field Load

The outbreak of fire has a great influence on. the carrying capacity of structure. the elastic modulus, flexural rigidity, yield strength and natural frequency of structural steel are discussed on the effects of fire temperature in this paper. The simply supported steel beam to form a plastic hinge when the ultimate moment changes in the first phase of the beam movement and the dynamic responses of simply supported steel beams under different loads were analyzed. Specific to the Q235 H-shaped steel case of H-shaped steel was calculated in high temperature structural steel bending stiffness, yield strength and natural frequency of simply supported steel beam in the two stages of movement of the dynamic response is analyzed. The analysis showed that the fire duration was 2.5h, the yield strength of steel is only 85.63N/mm2, a drop of 63.56%, while the component in the fire field in two phases dynamic deflection is also rapid growth, an increase of over 50%, almost lost capacity, in addition to members of this excessive plastic deformation is a serious cause of the damaged component. Dangerous. Therefore, in order to ensure the safety of steel structure design, the adverse effects should be taken into full consideration in the design of simply supported steel beams.

철도 인프라 적용 교량형 조립식 모듈의 경량화 설계

This paper describes a method to design a lightweight modular bridge for a railway infrastructure system. A lightweight design was achieved using the material selection method. Aluminum extrusions and honeycomb sandwich composites were selected as the best materials to reduce the weight of the upper structure of a conventional modular bridge made of carbon-steel material. The structural integrity of the lightweight modular bridge was evaluated under vertical and wind loads. The twisting and bending natural frequencies were also evaluated to investigate its dynamic characteristics. The results showed that the structural integrity and natural frequencies of the lightweight modular bridge, made of aluminum extrusion and sandwich composites, satisfied the design requirements. Moreover, it was found that the weight of the conventional modular bridge made of carbon steel could be reduced by a maximum of 47% using lightweight materials.

Airfoil-based electromagnetic energy harvester containing parallel array motion between moving coil and multi-pole magnets towards enhanced power density.

In this letter, the airfoil-based electromagnetic energy harvester containing parallel array motion between moving coil and trajectory matching multi-pole magnets was investigated. The magnets were aligned in an alternatively magnetized formation of 6 magnets to explore enhanced power density. In particular, the magnet array was positioned in parallel to the trajectory of the tip coil within its tip deflection span. The finite element simulations of the magnetic flux density and induced voltages at an open circuit condition were studied to find the maximum number of alternatively magnetized magnets that was required for the proposed energy harvester. Experimental results showed that the energy harvester with a pair of 6 alternatively magnetized linear magnet arrays was able to generate an induced voltage (Vo) of 20 V, with an open circuit condition, and 475 mW, under a 30 Ω optimal resistance load operating with the wind speed (U) at 7 m/s and a natural bending frequency of 3.54 Hz. Compared to the traditional electromagnetic energy harvester with a single magnet moving through a coil, the proposed energy harvester, containing multi-pole magnets and parallel array motion, enables the moving coil to accumulate a stronger magnetic flux in each period of the swinging motion. In addition to the comparison made with the airfoil-based piezoelectric energy harvester of the same size, our proposed electromagnetic energy harvester generates 11 times more power output, which is more suitable for high-power-density energy harvesting applications at regions with low environmental frequency.

Non-linear aeroelastic analysis in the time domain of high-aspect-ratio wings: Effect of chord and taper-ratio variation

ABSTRACTCommercial jets usually have relatively low-aspect-ratio wings, in spite of the associated benefits of increasing the wing aspect-ratio, such as higher lift-to-drag ratios and ranges. This is partially explained by the fact that the wing becomes more flexible by increasing the aspect-ratio that results in higher deflections which can cause aeroelastic instability problems such as flutter. An aeroelastic computational framework capable of evaluating the effects of geometric non-linearities on the aeroelastic performance of high-aspect-ratio wings has been developed and validated using numerical and experimental data. In this work, the aeroelastic performance of a base wing model with 20 m span and 1 m chord is analysed and the effect of changing the wing chord or the taper-ratio is determined. The non-linear static aeroelastic equilibrium solutions are compared in terms of drag polar, root bending moment and natural frequencies, and the change in the flutter speed boundary is assessed as a function of aspect-ratio using a time-marching approach.

Dynamic Analysis of Heavy Vehicle Medium Duty Drive Shaft Using Conventional and Composite Material

The main highlight of this study is structural and modal analysis of single piece drive shaft for selection of material. Drive shaft is used for torque carrying from vehicle transmission to rear wheel differential system. Heavy vehicle medium duty transmission drive shaft was selected as research object. Conventional materials (Steel SM45 C, Stainless Steel) and composite materials (HS carbon epoxy, E Glass Polyester Resin Composite) were selected for the analysis. Single piece composite material drive shaft has advantage over conventional two-piece steel drive shaft. It has higher specific strength, longer life, less weight, high critical speed and higher torque carrying capacity. The main criteria for drive shaft failure are strength and weight. Maximum modal frequency obtained is 919 Hz. Various harmful vibration modes (lateral vibration and torsional vibration) were identified and maximum deflection region was specified. For single-piece drive shaft the natural bending frequency should be higher because it is subjected to torsion and shear stress. Single piece drive shaft was modelled using Solid Edge and Pro-E. Finite Element Analysis was used for structural and modal analysis with actual running boundary condition like frictional support, torque and moment. FEA simulation results were validated with experimental literature results.

Vibration response analysis of blade-disk dovetail structure under blade tip rubbing condition

A cyclic sector corresponding to blade-disk structure with dovetail connection (1/38 blade-disk) is studied and the finite element (FE) model of this structure is established based on ANSYS software. A revised normal rubbing force model is developed and a pulse force model is established to simulate the local rubbing phenomenon between the blade and elastic casing based on the revised model. The effects of the rubbing under different rotating speeds and penetration depths on the blade vibration response and contact behaviors of dovetail interface are analyzed. The results show that the rubbing will cause amplitude amplification phenomenon when the multiple frequency components are close to the first bending and first torsion natural frequencies. The arch bending of the blade caused by blade-tip rubbing can be identified by evaluating the displacement and stress of the blade in the radial direction (y-direction). The dynamic stress in the process of rubbing gradually changes from alternation between tension and compression stress to the tension stress with the increasing rotating speed. Maximum contact sliding distance may change dramatically when the rubbing force is greater than the centrifugal force. With the increase of rotating speed, the contact pressure increases under the centrifugal force and its fluctuation under rubbing is smaller at higher rotating speeds.

An analytical method to estimate the natural bending frequency of the spans of railway through truss bridges with steel-and-concrete composite decks

The Eurocode 1 standard requires railway bridges to be checked against the hazard of resonance. To perform this task, the first natural bending frequency of a bridge span has to be computed. The analytical method based on the single-beam analogy is sufficiently accurate for simply supported beam-girder bridges. This paper proves the applicability of the method for through truss bridge spans with steel-and-concrete composite decks; this is despite the fact that the spans do not strictly comply with the assumptions of this method. The flexural stiffness of the bridge span is computed taking into account the joint action of the truss girders and the composite deck; the limited extent of span shear stiffness due to diagonal bracing is also considered. A uniform distribution of the span mass over the span length is assumed. The mass is computed based on geometrical characteristics of truss and deck members, taking into account gusset plates, stiffeners, connectors and fittings. The analytical method and finite element method are applied to find the natural frequencies of two through truss bridge spans with composite decks. Computed results are compared with frequencies recorded during bridge testing. Taking into consideration the joint action of the truss girders and the composite deck, as well as the limited shear stiffness provided by diagonal bracing, significantly improves the accuracy of the assessment provided by the analytical method. The natural bending frequencies computed for the two bridge spans fall within the range of 0.95–1.05 of the recorded values.