Clamped-free Boundary Conditions Research Articles

LVI analysis on the beams with nonuniform cross-sectional area using exponential shear–strain function

Purpose The study aims to investigate the low-velocity impact (LVI) on the surface of a beam with a changeable cross-sectional area. In the study “LVI on a beam with a changeable cross-sectional area and clamped-free boundary conditions”, the effect of changes in the cross-section are on the contact force, the beam displacement, the impactor displacement and the impactor velocity are investigated. Design/methodology/approach To obtain the motion equations, first, a field of displacements of the beam is written using third-order shear deformation of beams, including the exponential shear–strain function, and then the energy method is used. By combining Hamilton’s approaches and Ritz’s method, finally, the equations of motion are extracted. Using ABAQUS finite element code, validation of the theoretical approach is carried out. In this study, the beam with changeable cross-sectional area is considered in such a way that the height of the beam is constant, but the width of the beam changes linearly. Findings The results show that assuming the width of the beam in the clamped support is constant, an increase in the width of the beam in the free support leads to an increase in the peak contact force and the residual velocity of the impactor, also, the peak displacement of the beam and the impactor are decreased. Originality/value It can be shown from the analysis of LVI on beams with nonuniform cross-sectional area that the important influence on the contact force, impactor residual velocity, beam displacement and impactor displacement is achieved.

Vibration and buckling analysis of porous metal foam thin-walled beams with closed section

This paper investigates the vibration and buckling analysis of porous metal foam thin-walled box beams. These beams exhibit a unique structural configuration with symmetrical and asymmetrical porosity distributions along their wall thickness, thereby altering the effective mechanical properties. The first-order shear deformable beam theory is employed and the governing equations are derived using the Hamilton’s principle. Numerical results are presented for porous metal foam thin-walled box beams under simply-supported, clamped–clamped and clamped-free boundary conditions. The effects of various porosity parameters, length-to-side and side-to-wall-thickness ratios on the beams’ performance are also examined. A comprehensive comparison between the porous metal foam thin-walled box beams and their counterparts in the form of equivalent homogeneous are presented.

Novel Kuhn-Tucker conditions for vibration analysis in a functionally graded porous beam using the R-program

Functionally graded materials provide a flexible and individualized strategy for material design, allowing for optimization of properties and performance for particular purposes. The investigation considers the effects of simply supported (SS), clamped-clamped (CC) and clamped-free (CF) configurations. The study examines the vibration characteristics of bi-directional functionally graded porous beams (BDFGPB) using the third-order shear deformation theory, considering both even and uneven porosity conditions. The Hamilton method is used to derive equilibrium equations for beams, which are then solved using the Kuhn-Tucker technique and R-program. The BDFGPB's validity was verified by comparing it with open literature, revealing deviations of 3.19%, 1.25%, and 2.15% in non-dimensional natural frequency for SS, CC and CF boundary conditions. Furthermore, as the porosity index increases, the dimensionless natural frequency decreases, reducing beam stiffness and rigidity. This study demonstrates that porosity plays a critical role in the design of modern structures, as its ratio greatly impacts their performance and responsiveness.

DFT-based finite element analysis of compressive response in armchair phosphorene nanotubes

In this paper, the finite element method is utilized to evaluate the behavior of the armchair phosphorene nanotubes under the compressive loading. The energy equations of the molecular and structural mechanics are used to obtain the elemental properties. The critical compressive forces of various armchair phosphorene nanotubes are computed with clamped-clamped and clamped-free boundary conditions. Results show that the stability of armchair phosphorene nanotubes increases with increasing nanotube aspect ratio, particularly under clamped-clamped boundary conditions. Finally, the buckling mode shapes of armchair phosphorene nanotubes under different boundary conditions are compared. Our work offers valuable insights into how these nanotubes respond to mechanical stress, helps determine elemental properties, and investigates the effects of nanotube geometry and different boundary conditions on their stability. This knowledge has broad applications in fields like nanotechnology, materials science, and nanomechanics, advancing the understanding of nanoscale materials and their potential for various practical uses.

Experimental investigation of compressive behavior and vibration properties of layered hybrid foam formed by aluminum foam/EPS-filled syntactic foam

Hybrid foams are a type of composite material created by combining two foam materials. They are preferred in many applications due to their lightweight, high strength, and ability to absorb more energy. In this research, a new hybrid foam was designed for use in sandwich cores of structural materials. The hybrid foam was formed by combining closed-cell aluminum foam and expanded polystyrene (EPS)-filled syntactic foam. The EPS-filled syntactic foams were produced with the conventional mold casting technique. Uniaxial compressive behaviors (0.5 mm/min) of layered hybrid foams consisting of EPS-filled syntactic foam with three different densities and closed-cell aluminum foam were investigated experimentally and compared with conventional single-foamed materials. These results exhibited that in general, layered hybrid foams outperform conventional single-foamed materials in terms of compressive strength. Moreover, the natural frequency and damping ratio of the layered hybrid foams were investigated by vibration tests under clamped-free and free-free boundary conditions and compared to conventional single-foamed materials. It was established that the vibration damping capacity of the layered hybrid foams improved compared to the closed-cell aluminum foam. Additionally, the microstructure of the conventional single-foamed materials was examined by SEM. In the outcome of the research, the experimental results showed that layered hybrid foam provides an opportunity to design lightweight cellular materials with effective mechanical properties.

The effect of manufacture process on material properties of laminated composite cylinders using non-destructive technique

This paper presents a comparative investigation of the effect of hand lay-up to manufacture of composite structure on the elastic constants of composite cylindrical shells. Non-destructive technique for detection of elastic constants in laminated composite cylinders were applied. This technique based on numerical-experimental method that included experimental testing and numerical modelling. Experimental testing includes determination of natural frequencies in composite cylinders fixed at the ends. Composite construction fixed by upper end on the special test bench to realize clamped-free boundary conditions. 48 piezo sensors were glued on the outer surface of composite shells to measure strains during testing. Data received during experiments were converted in modal parameters using ARTEMIS software. By minimizing the error functional between the numerical and experimental data of the natural frequencies, the properties of laminated composite materials were obtained. It can be seen that elastic constants of the five composite shells have insignificant differences in values and be successfully applied for numerical calculation in further.

A Numerical Study on the Large Displacement in Functionally Graded Beam under Thermal Effect

The large displacement behavior of a Functionally Graded (FG) beam under uniform thermal load is investigated numerically. Six different effects are taken into consideration when examining the large displacement behavior of the beam. These are the effects of temperature, material, geometry, slenderness, force, and boundary condition. The nonlinear numerical analysis is carried out by using the Simulation mode of SolidWorks, which is a finite element-based commercial program. As a result, the displacement of the end of the beam increase with increasing temperature, slenderness ratio, and force. However, it decreases with increase in the ratio of ceramic in Functionally Graded Material (FGM) and the width of the beam. Moreover, as expected, the maximum and minimum displacements are obtained in the beams with Clamped-Free and Clamped-Clamped boundary conditions, respectively.

Dynamic Behavior Modeling of Natural-Rubber/Polybutadiene-Rubber-Based Hybrid Magnetorheological Elastomer Sandwich Composite Structures.

This study investigates the dynamic characteristics of natural rubber (NR)/polybutadiene rubber (PBR)-based hybrid magnetorheological elastomer (MRE) sandwich composite beams through numerical simulations and finite element analysis, employing Reddy's third-order shear deformation theory. Four distinct hybrid MRE sandwich configurations were examined. The validity of finite element simulations was confirmed by comparing them with results from magnetorheological (MR)-fluid-based composites. Further, parametric analysis explored the influence of magnetic field intensity, boundary conditions, ply orientation, and core thickness on beam vibration responses. The results reveal a notable 10.4% enhancement in natural frequencies in SC4-based beams under a 600 mT magnetic field with clamped-free boundary conditions, attributed to the increased PBR content in MR elastomer cores. However, higher magnetic field intensities result in slight frequency decrements due to filler particle agglomeration. Additionally, augmenting magnetic field intensity and magnetorheological content under clamped-free conditions improves the loss factor by from 66% to 136%, presenting promising prospects for advanced applications. This research contributes to a comprehensive understanding of dynamic behavior and performance enhancement in hybrid MRE sandwich composites, with significant implications for engineering applications. Furthermore, this investigation provides valuable insights into the intricate interplay between magnetic field effects, composite architecture, and vibration response.

Designing regression approach for mode frequency prediction of empty and filled steel silos

In this study the primary objective is to design prediction model for the free vibration analysis of thin circular cylindrical steel silos having various aspect ratios in empty and varying filled conditions for different types of closures. A finite element method (FEM) is used to carry out the free vibration analysis of steel silos. It is found that the effect of different aspect ratios slender, intermediate slender and squat steel silos is very significant for dynamic response of silo. The silos are considered having open, flat and cone type of closures at its top end. The clamped-free (cantilever) boundary condition is taken for this approach as actual silos are fixed at flat-base. The structural mass of thin cylindrical shell steel silo is made constant for a particular height and diameter. The eigenvalues of the thin cylindrical shell steel silos are extracted by using block Lanczos method. The free vibrations of thin cylindrical shell steel silo with different aspect ratios, radius to thickness ratio are studied. From the present studies it is seen that as aspect ratio increases the fundamental frequency is reduced in empty silo. It is more in the case of squat silo. It can be seen that the fundamental frequency is less in the case of flat closure in all the aspect ratios of the silo. The frequency values are more in the case of cone closure is observed. Also as the mode number increases the modal frequency value increases. Further, as the filling level is increased the modal frequency also increases. Finally, regression approach is adopted for predicting the mode frequency of empty and filled silos for wide range of aspect ratios.

Investigation of vibration of the nano rotating blade coupled with viscous fluid medium by considering the nonlocal elastic theory

In view of the great importance of dynamical behavior prediction of nanostructures in contact with fluid and their vast range of applications in biomedical engineering, aerospace, etc., in this research, the free vibration of a Nanoscale Euler-Bernoulli rotating beam coupled with incompressible viscous fluid is studied. Small-scale effects are applied by using nonlocal elasticity theory. Using the Navier-Stokes relation, the interaction forces between the fluid and nanobeam are obtained. Governing differential equations have been solved by Galerkin method and the system vibrations frequency response has been obtained for clamped-free boundary condition. Based on the results of this research, nonlocal elasticity has a different effect on different vibration modes. The frequency of the nanobeam coupled with the fluid quickly increases when applying this theory, and the presence of fluid reduces the natural frequencies.

Analytical modelling of a multifunctional heterogeneous beam-bending analysis

Pores have a crucial role in shaping the features of functionally graded materials. If the distribution of pores is permitted to gradually rise from the outside inwards towards the center, many more features can be incorporated. A shear shape function is used to analyze the bending response of a two-directional functionally graded porous beam (FGPB), which can be considered a multifunctional heterogeneous material while accounting for both even as well as uneven porosity distributions. The power law is adapted to modify the material properties of FGPB through out the length and thickness dimensions. The notion of virtual displacements is applied when establishing equilibrium equations in FGPB. The Navier method is applied to clamped-clamped, clamped-free boundary conditions and simply supported boundary conditions to offer solutions to FGPB. The proposed methodology is supported by the numerical results of functionally graded beams from earlier studies. The dimensionless deflections and stresses under study are affected by gradient exponents, porosities, and aspect ratios.

Non-linear dynamics of cantilevered circular cylindrical shells with thickness stretch, containing quiescent fluid with small-amplitude sloshing

Studies on nonlinear vibrations of circular cylindrical shells containing fluid are mostly limited to thin simply supported shells, such that no results regarding the behavior of thick cantilevered shells with shear and thickness deformations can be found in the literature. In this article, for the first time, thin and thick circular cylindrical shells with clamped-free boundary conditions are modeled according to the third-order shear deformation theory with thickness stretch. The contained liquid satisfies the Laplace equation and linearized boundary conditions are imposed at the free surface and at the contact with the shell. Using Lagrange equations, the governing differential equations of the system in terms of shell-fluid interaction are derived. In linear free vibration analysis, it was found that increasing the fluid level inside the shell and decreasing the shell thickness, cause a significant rise in the fluid free-surface wave elevation. This increase is such that limits the application of linear sloshing theory. It was observed that the fluid presence can change the nonlinear behavior from softening to hardening type and intensifies the shell thickness deformation. In addition, the presence of contained liquid reduces the circumferential dynamic contraction of the shell caused by large amplitude vibrations.

A strain rate-dependent analytical approach for low-velocity impact on the beam composed of silicon-nitride and stainless-steel

Purpose The purpose of this study is to investigate the strain rate effect on the problem of low-velocity impact (LVI) on a beam, including silicon nitride and stainless steel materials. Design/methodology/approach Based on the nonlinear Hertz impact mechanism, the energies related to the impactor and the beam are written, and motion equations are derived using the Lagrangian mechanics and Ritz method. The strain rate term is represented as a damping matrix in the equations of motion. In the issue of LVI on the silicon nitride and stainless steel beam, the effect of internal viscous damping coefficient in simply–simply and clamped–free boundary conditions are studied. Also, the influence of the volume fraction index in the range between zero and one and greater than one on the impact response is investigated. Findings The results make it clear that the strain rate parameter had little effect on the response in LVI. Also, an increase in the volume fraction index has led to a decrease in the contact force and an increase in the rebound velocity of the impactor. Originality/value The effect of strain rate on LVI is theoretically studied in this paper, while in most of the papers, this effect is investigated experimentally and numerically.

Novel Approach-Based Sparsity for Damage Localization in Functionally Graded Material

Model-based approaches have been widely employed in damage detection and localization studies. However, alternative techniques, such as built-in online detection methods, hold promise for future advancements in structural health monitoring technologies. In this research paper, we present a dynamic algorithm specifically designed for accurate damage localization in functionally graded plates. The suggested method involves the creation of a grid matrix that captures the dynamic response of the structure over time. Subsequently, an optimization process is performed using a linear equation that incorporates the information contained within the grid, enabling the precise localization of damage. To address the inherent sparsity of the localization nature, we utilize the FISTA (fast iterative shrinkage-thresholding algorithm) as a problem solver. The effectiveness of our approach is evaluated through experimental tests on a functionally graded plate with clamped free boundary conditions. Multiple damage scenarios are investigated, including cases with damage signals on and off-the-grid. The results demonstrate that our proposed approach is capable of accurately predicting the position of damage, indicating its suitability for application in low-size data systems.

Nonlocal boundaries and paradoxes in thermoelastic vibrations of functionally graded Non-Uniform cantilever nanobeams and annular nanoplates

The paradoxical behaviour of free vibrations of cantilever nanostructures using Eringen's nonlocal elasticity model is always a debatable issue for researchers in the past few years. According to this, the frequency of cantilever nanobeams increases with the increase in the value of the nonlocal parameter while it decreases for other boundary conditions. In thermoelastic vibrations, the frequency of nanobeams increases and decreases with the increase in temperature at the surfaces for clamped-free and other boundary conditions, respectively. Therefore, investigation of these structures is of critical importance. In view of this, the authors have proposed the modified boundary conditions for the well-established differential model of Eringen’s nonlocal elasticity theory to deal with the free edges of the structural components. Using these conditions, the vibration analysis of nonuniform functionally graded nanobeams and axisymmetric annular nanoplates is presented for clamped-free boundary conditions on the basis of first-order shear deformation theory and physical neutral plane. The thickness of these structural elements (nanobeams and annular nanoplates) is assumed to be varying linearly and parabolically in the axial direction. The temperature-dependent mechanical properties of the material are obtained by the power law. The nonlocal governing equations and modified boundary conditions for the structural elements are discretized using the generalized differential quadrature method and solved by MATLAB to obtain the fundamental frequency. For the present modifications, the behaviour of frequency parameter for different values of nonlocal parameter, volume fraction index, temperature difference, thickness parameter, and taper parameter is discussed.

Flügge shell theory and solution for the vibration analysis of a thin elastic cylindrical panel of varying curvature

In this paper, the free vibration behavior of isotropic and orthotropic cylindrical shell panels with variable curvature subjected to certain ends boundary conditions is analyzed by the transfer matrix technique and solved by the Romberg integration approach. Flügge’s shell theory is employed to model the governing three-dimensional equations of vibration and Fourier’s approach is used to deform the displacement fields as trigonometric functions in the longitudinal direction. As a result of a semi-analytical procedure is conducted, the coupled governing equations of panel problem can be written in a matrix partial differential equation of first order with variable coefficients and reduced from the two-dimensional problem to one-dimensional one which is solved numerically as an initial-value problem. The proposed model is applied to get the natural frequencies and mode shapes for the vibration behavior of two-side simply-supported cylindrical panels with simply-supported, free-free, clamped-free and clamped-clamped ends boundary conditions lying at their curved edges. The sensitivity of the vibration behavior to the variety of curvature, panel parameters and material orthotropy of shell is studied.

Effect of Axial Load and Thermal Heating on Dynamic Characteristics of Axially Moving Timoshenko Beam

A thermoelastic coupling model of axially moving beam based on Timoshenko beam model is established, which comprehensively takes into account the length–diameter ratio, axial load and moving speed, and unify the axial tension and compression load into one governing equation, and study the influence of the length–diameter ratio, axial load, motion speed and thermal load of the beam model. The differential governing equations of transverse vibration of axially moving beam with considering the axial tensile and compressive loads are established based on the Timoshenko beam theory and Hamilton’s principle. The dynamic characteristics of different slender beams with axial load and pinned–pinned, clamped–clamped and clamped-free boundary conditions are investigated, respectively. The dimensionless frequencies of beam calculated numerically with the differential quadrature method (DQM) are compared with analytical solutions for some special cases. For axially moving beams working in the thermal environment, the temperature field inside the beam is simulated by the heat conduction equations, and the thermal effect on dynamic characteristics of beam is studied under different thermal heating cases.

Buckling and free vibrations behaviour through differential quadrature method for foamed composites

The current work focuses on predicting the buckling and free vibration frequencies (fn) of cenosphere reinforced epoxy based syntactic foam beam under varying loads. Critical buckling loads (Ncr) and fn are predicted using the differential quadrature method (DQM). Ncr and fn have been calculated for beams of varying cenosphere volume fractions subjected to axial load under clamped-clamped (CC), clamped-simply (CS), simply-simply (SS), and clamped-free (CF) boundary conditions (BC′s). Upon increasing the cenosphere volume fraction, Ncr and fn of syntactic foam composites increases. These numerical outcomes are compared with the theoretical values evaluated through the Euler-Bernoulli hypothesis and further compared with experimental outcomes. Results are observed to be in precise agreement. The results of the DQM numerical analysis are given out for the different BC′s, aspect ratios, cenosphere volume fractions, and varying loads. It is perceived that depending on the BC′s, the type of axial varying loads and aspect ratios has a substantial effect on the Ncr and fn behaviour of the syntactic foam beams. A comparative study of the obtained results showed that the beam subjected to parabolic load under CC boundary conditions exhibited a higher buckling load.

A novel sensor design for a cantilevered Mead-Marcus-type sandwich beam model by the order-reduction technique

red A novel space-discretized Finite Differences-based model reduction introduced in [J. Liu and B.Z. Guo, SIAM J. Control Optim. 58 (2020) 2256-228] is extended to the partial differential equations (PDE) model of a multi- layer Mead-Marcus-type sandwich beam with clamped-free boundary conditions. The PDE model describes transverse vibrations for a sandwich beam whose alternating outer elastic layers constrain viscoelastic core layers, which allow transverse shear. The major goal of this project is to design a single tip velocity sensor to control the overall dynamics on the beam. Since the spectrum of the PDE cannot be constructed analytically, the so-called multipliers approach is adopted to prove that the PDE model is exactly observable with sub-optimal observation time. Next, the PDE model is reduced by the “order-reduced” Finite-Differences technique. This method does not require any type of filtering though the exact observability as h → 0 is achieved by a constraint on the material constants. The main challenge here is the strong coupling of the shear dynamics of the middle layer with overall bending dynamics. This complicates the absorption of coupling terms in the discrete energy estimates. This is sharply different from a single-layer (Euler-Bernoulli) beam.

폭이 테이퍼진 레일레이 외팔보의 진동 해석

The vibration characteristics of width-tapered Rayleigh beams with the clamped-free boundary condition is investigated, and subsequently used to analyze the effect of rotary inertia and taper ratio on the natural frequencies and mode shapes of the beams. The width of the cross-section is assumed to linearly reduce along the length of the beam. The frequency determinant is obtained by using the transfer matrix, and the differential equation is solved using the power series. The accuracy of the method is demonstrated by comparing the results obtained in this study with those obtained in the previous study. In addition, the effect of rotary inertia and taper ratio on the eigen-pairs of width-tapered Rayleigh beams with respect to the variation of the total length is investigated.