Simply Supported-simply Supported Research Articles

Chebyshev–Ritz and Navier's methods for hygro‐magneto vibration of Euler–Bernoulli nanobeam resting on Winkler–Pasternak elastic foundation

AbstractThis research employs Chebyshev–Ritz method along with Navier's method to investigate the vibration characteristics of a nanobeam subject to a longitudinal magnetic field and linear hygroscopic environment. The nanobeam is characterized by a Winkler–Pasternak elastic foundation and follows the nonlocal Euler–Bernoulli beam theory. The governing equation of motion is derived using Hamilton's principle, and non‐dimensional frequency parameters are computed for Simply Supported‐Simply Supported (SS), Clamped‐Clamped (CC), and Clamped‐Free (CF) boundary conditions. The motivation behind this study is to provide a comprehensive and efficient analytical framework for understanding the dynamic behavior of nanobeams in complex environments. By investigating the influence of magnetic and hygroscopic factors on the vibration characteristics of nanobeams, this research aims to offer valuable insights for the design and optimization of nanoscale structures. Employing shifted Chebyshev polynomials as shape functions in Chebyshev–Ritz method offers several advantages in the proposed model. Firstly, these polynomials possess orthogonal properties, which can significantly enhance computational efficiency. The orthogonality of shifted Chebyshev polynomials allow for simpler and more streamlined numerical computations compared to non‐orthogonal basis functions. Additionally, the orthogonality ensures that the resulting system of equations is well‐conditioned, even for higher‐order polynomial approximations. A closed‐form solution for SS boundary condition is obtained through Navier's method. Convergence analysis is performed to validate the accuracy and effectiveness of the proposed model against existing models. The non‐dimensional frequency parameters obtained using both Navier's method and Chebyshev–Ritz method demonstrate strong agreement, further validating the proposed nanobeam model. Additionally, a comprehensive parametric study evaluates the impact of various characteristics, including the small‐scale parameter, Winkler modulus, shear modulus, magnetic parameter, and hygroscopic parameter. The findings contribute to a nuanced understanding of nanobeam vibrations under the influence of a magnetic field and hygroscopic environment, providing valuable insights for the design and optimization of nanoscale structures in practical applications.

Analysis of thermoelastic damping and modeling of piezothermoelastic plate resonators with voids

A comprehensive micro-electro mechanical systems (MEMS) model for analysis of a smart composite piezothermoelastic thin plate with voids, in context of Lord-Shulman theory of thermoelasticity has been developed. The work culminates in the derivation of the closed form analytic expressions for thermoelastic damping, frequency shift, electric potential change, volume fraction change, and displacements. Some numerical illustrations and graphical interpretations of analytical developments in case of PZT-5A material piezothermoelastic plate under clamped-simply supported (CS) and simply supported-simply supported (SS) boundary conditions have been presented with the help of MATLAB programming software.

Frequency shifts and thermoelastic damping in distinct Micro-/Nano-scale piezothermoelastic fiber-reinforced composite beams under three heat conduction models

The present work aims to fulfill two primary objectives. Firstly, to present the micro-mechanics model of piezothermoelastic fiber-reinforced composite (PTFRC) medium using analytical techniques, viz., Strength of Materials (SM) & Rule of Mixtures (RM) and graphically establish some of its electro-mechanical-thermal advantages over monolithic PT materials. Secondly, to analytically study the thermoelastic damping (TED) and frequency shift (FS) in micro-/nano-scale PTFRC beams. TED and FS are analyzed considering the linear Euler-Bernoulli theory when the beam is under four distinct boundary conditions, viz., clamped-clamped (CC), simply supported-simply supported (SS), clamped-free (CF), and clamped-simply supported (CS). The eigenvalues of each beam are calculated numerically with 20 iterations each using the Newton-Raphson technique, followed by the analysis of TED and FS considering the assumed boundaries. The Classical dynamical coupled (CL), Lord-Shulman (LS), and Green-Lindsay (GL) thermoelasticity theories are exclusively analyzed. The effects of thermal relaxation parameters, fiber volume fraction, micro and nano beam dimensions, and the first two modes (M1 & M2) on the TED and FS are graphically illustrated. The critical thickness (CrTh) and critical length (CrLt) of all four beams influenced by the existing parameters are scrupulously investigated. The robustness of the obtained results is validated by matching them with previous literature. For high Ωf, the PTFRC beam has superior electro-mechanical-thermal properties and greater Q-factors compared to monolithic beam. However, the Q-factor increases as Ωf decreases. This facilitates the choice of a suitable value of Ωf for designing and optimizing frequency-sensitive microelectromechanical (MEM) and nanoelectromechanical (NEM) PTFRC beams.

Free vibration and buckling characteristics of porous functionally graded materials (FGMs) micro‐beams based on the modified couple stress theory

AbstractBased on the modified couple stress theory (MCST), the free vibration and buckling characteristics of porous functionally graded materials micro‐beams (P‐FGMs and E‐FGMs) are studied. Firstly, two types of pore distribution are considered for functionally graded materials (FGMs) micro‐beams. The governing differential equations of free vibration and buckling of FGMs micro‐beams are derived by Hamilton's principle and dimensionless. Then, the dimensionless governing differential equations and their boundary conditions are transformed by differential transformation method (DTM), then the equivalent algebraic characteristic equations with natural frequencies are obtained. Finally, the dimensionless fundamental frequency and critical buckling load of the free vibration of the porous FGMs micro‐beam under four different boundary conditions of clamped‐clamped (C‐C), clamped‐simply supported (C‐S), simply supported‐simply supported (S‐S) and clamped‐free (C‐F) are calculated. The results are compared with the existing literature data, and the correctness is verified. The effects of porosity, scale parameter, stiffness ratio, axial load and gradient index on the dimensionless natural frequency and critical buckling load of FGMs micro‐beams are discussed.

Frequency Analysis for Functionally Graded Material Cylindrical Shells: A Significant Case Study

Cylindrical shells play an important role for the construction of functionally graded materials (FGMs). Functionally graded materials are valuable in order to develop durable materials. They are made of two or more materials such as nickel, stainless steel, zirconia, and alumina. They are extremely beneficial for the manufacturing of structural elements. Functionally graded materials are broadly used in several fields such as chemistry, biomedicine, optics, and electronics. In the present research, vibrations of natural frequencies are investigated for different layered cylindrical shells, those constructed from FGMs. The behavior of shell vibration is based on different parameters of geometrical material. The problem of the shell is expressed from the constitutive relations of strain and stress with displacement, as well as it is adopted from Love’s shell theory. Vibrations of natural frequencies (NFs) are calculated for simply supported-simply supported (SS-SS) and clamped-free (C-F) edge conditions. The Rayleigh–Ritz technique is employed to obtain the shell frequency equation. The shell equation is solved by MATLAB software.

Frequency shifts and thermoelastic damping in different types of Nano-/Micro-scale beams with sandiness and voids under three thermoelasticity theories

The present work focuses on the analytical study of thermoelastic damping (TED) and frequency shift (FS) in micro-scale beams and TED in nano-scale beams. The considered thermoelastic beam is homogeneous, isotropic, and consists of sandiness and voids (TESV). TED and FS of the beam are studied under four types of distinct boundary conditions, viz. clamped-clamped (CC), simply supported-simply supported (SS), Cantilever, and clamped-simply supported (CSS), considering different types of practical engineering applications. Newton–Raphson method is used to numerically evaluate the first two eigenvalues of each type of beam. The expressions of the characteristic roots are also provided. Based on the linear Euler–Bernoulli theory, the closed-form expressions for the beams’ transverse vibrations are derived. Analytical expressions for the deflection, change in temperature, and void volume fraction are also obtained, followed by TED and FS analysis for the considered boundaries. Three theories of thermoelasticity, viz. the Classical dynamical coupled (CL), Lord–Shulman (LS), and Green–Lindsay (GL) thermoelasticity theories are considered in this problem. The influences of thermal relaxation times owing to the three thermoelasticity theories, voids, sandiness, micro & nano dimensions of the beam, and the first two modes (M1 & M2) on the TED and FS are illustrated graphically. Critical thickness (CrTh) and critical length (CrLt) of all four beams under the influence of the existing parameters are meticulously analyzed. Some special cases of this problem are discussed, and the obtained outcomes are validated with well-established results found in the extant literature. In-depth analysis of the effects of thermoelastic coupling and other existing parameters on the natural frequency of micro-/nano-scale resonators, among other applications, is quintessential for designing and optimizing frequency-sensitive nanoelectromechanical systems (NEMS) and microelectromechanical systems (MEMS). Hence, the present analysis may have significant application and contribution in areas utilizing NEMS/MEMS devices.

Semi-Analytical Solution of In-Plane Vibrations of Circular Arches Carrying Added Point Masses

In spite of the current use in modern civil engineering of straight structural components made of reinforced concrete, such as beams and slabs, arches still remain quite often adopted in architectural design, due to their interesting mechanical and esthetical properties. Therefore, practical analytical and numerical tools should be available in order to allow designers to ensure the resistance and stability of this type of structures, especially in the dynamic regime and when point masses are added. The sixth degree differential equation of motion of inextensible arches is known to be quite difficult to deal with and is not applicable to the case examined here of arches with added point masses [1]. The procedure adopted in this article consisted on finding relatively easy particular solutions of this equation satisfying the end conditions and using them as trial functions in the Rayleigh-Ritz formulations of the practical cases of interest. Circular arches of different opening angles are examined here under the classical hypotheses: (1) the effect of shear deformation and rotary inertia are neglected (2) the arch axis is inextensible (3) the dimensions of the cross-section are small in comparisons with the radius;(4) the cross-section is uniform. Three different kinds of end conditions are considered: simply supported-simply supported (SS), clamped-simply supported (CS) and clamped-clamped (CC). The Rayleigh-Ritz method allowed in each case to determine, via an algebraic solution of the associated eigenvalue problem, the first seven mode shapes and natural frequencies of arches with different values of the opening angle. The comparison with the results available in the bibliography was satisfactory and the mode shapes were plotted. The second part of this work was devoted to arches with added concentrated masses: likewise, the numerical values found for the natural frequencies compare quite well with the few references available and the mode shapes are plotted showing the effects of the opening angle, the mass and the positions of the added point masses.

A novel fractional nonlocal model and its application in buckling analysis of Euler-Bernoulli nanobeam

The present study aims to analyze the buckling behavior of Euler–Bernoulli nanobeam in conjunction with a novel fractional nonlocal model namely conformable fractional nonlocal model. Fractional models are getting more popular among the researchers because of its applicability and fixability to handle many complex physical phenomena which are not possible to model with integer operators. Also, the main advantage of fractional models over integer model is its applicability to handle both the integer and noninteger operators which makes it much more flexible in term of real-world application. In this regards, the nonlocal constitutive relation is developed in conjunction with conformable fractional derivatives and fractional strain energy to analyze the buckling behavior of Euler–Bernoulli Nanobeam. In this study, the Simply Supported-Simply Supported (SS), Clamped-Simply Supported, and Clamped-Clamped boundary conditions are taken into the investigation with the help of the Differential Quadrature Method (DQM). Critical buckling load parameters are computed for the SS, CS, and CC boundary conditions from generalized eigenvalue problem. Graphical, as well as tabular results, are calculated by using MATLAB programmes and effects of various parameters such as fractional parameter, nonlocal parameter, aspect ratio on critical buckling load parameters extensively studied.

A numerical investigation on vibration analysis of fiber reinforced and truncated conical hollow shells with different fiber orientations

Truncated conical shells have been extensively used in a wide range of engineering applications dueto their special geometric shapes, and their vibration properties have been interested by many researchers in recentyears. In this study, vibration analysis of truncated conical hollow shells constructed from the fiber reinforcedcarbon/epoxy composites was investigated for different fiber orientation angles ([0o/90o], [15o/-75o], [30o/-60o], and[45o/-45o]). Finite element analysis (FEA) was performed by using ABAQUS software for prediction of naturalfrequencies and mode shapes within the extended frequency range. Natural frequency values were determinedfor constant semi-vertex angle of the conical structure (45o), and boundary conditions of clamped-clamped(C-C), simply supported-simply supported (S-S) and clamped-free (C-F). It is concluded that natural frequency hasbeen significantly affected by boundary conditions while it has been little effected by fiber orientation angle thoseeffecting on mode shapes associated with natural frequencies.

Finite Element Analysis of Thin Circular Cylindrical Shells

Free vibration finite element (FE) analysis of thin circular cylindrical shells is investigated. The circular cylindrical shell can vibrate in different modes and theoretically infinite modes are possible. The axial mode, m and circumferential mode, n, in any of their combinations define the modes and the corresponding modal frequencies. The shell elements are used to model the thin circular cylindrical shells. The eigenvalues of the shell are extracted using block Lanczos iteration method. Detailed mesh convergence studies are performed for different height or length to radius ( H/R) ratios. Importantly, selection of appropriate FE mesh size criteria based on the perimeter and height of the circular cylindrical shell, as well as thickness to radius (h/R) ratio are shown for various boundary conditions. The modal frequencies of the cylindrical shell are investigated for different boundary conditions such as clamped-clamped (C-C), clamped-free (C-F), and simply-supported - simplysupported (S-S). The effects of height to radius ( H/R) ratio and thickness to radius ( h/R) ratio on the modal frequencies of the cylindrical shells are also studied. For all the considered boundary conditions, the modal frequencies of the cylindrical shells increase with higher circumferential mode number and also with the increase in the h/R ratio. The modal frequencies are observed to be the lowest in the case of the C-F boundary condition of the shell.

Free transverse vibration analysis of size dependent Timoshenko FG cracked nanobeams resting on elastic medium

In the present study, free transverse vibration of a cracked functionally graded (FG) size dependent Timoshenko nanobeam which is resting on polymer elastic foundation is investigated. It is supposed that the material properties of the FG nanobeam are varying continuously across the thickness according to the power-law distribution. To considering the small scale effect, the Eringen's nonlocal theory is used and for accounting the effect of polymer elastic foundation, the Winkler model is proposed. For this purpose, the equations of motion of the FG Timoshenko nanobeam and boundary conditions are obtained by using Hamilton's principle. To find the analytical solutions for equations of motion of FG nanobeam, the separation of variable method is employed. Two cases of boundary conditions i.e., simply supported-simply supported (SS) and clamped---clamped (CC) are investigated in the present work. Numerical results are demonstrating the good agreement between the results of present article and some cases available in the literature. The emphasis of the present study is based on investigating the effect of various parameters such as crack severity, crack position, gradient index, mode number, nonlocal parameter, elastic foundation parameter and nanobeam length. It is clearly revealed that the vibrational behavior of a FG nanobeam is significantly depending on these effects. Also, these numerical results can be serving as benchmarks for future studies of FG nanobeams.

Thermo-electro-mechanical vibration of postbuckled piezoelectric Timoshenko nanobeams based on the nonlocal elasticity theory

In this study, free vibration behavior of piezoelectric Timoshenko nanobeams in the vicinity of postbuckling domain is investigated based on the nonlocal elasticity theory. It is assumed that the piezoelectric nanobeam is subjected to an axial compression force, an applied voltage and a uniform temperature change. Using Hamilton principle, the governing differential equations of motion incorporating von Kármán geometric nonlinearity and the corresponding boundary conditions are derived and then discretized on the basis of generalized differential quadrature (GDQ) scheme. After solving the parameterized equations using Newton–Raphson technique, a dynamic analysis based on a numerical solution strategy is performed to predict the natural frequencies of piezoelectric nanobeams associated with both prebuckling and postbuckling domains. Numerical results are presented to study the effects of nonlocal parameter, temperature rise and external electric voltage on the size-dependent vibration behavior of piezoelectric nanobeams with clamped–clamped (C–C), clamped-simply supported (C-SS) and simply supported-simply supported (SS-SS) end conditions. It is demonstrated that these parameters may shift the postbuckling domain to higher or lower applied axial loads.

Free Vibration Analysis of Beams Considering Different Geometric Characteristics and Boundary Conditions

In this study, free vibration of square cross-sectioned aluminum beams is investigated analytically and numerically under four different boundary conditions: Clamped-Clamped (C-C), Clamped-Free (C-F), Clamped-Simply Supported (C-SS) and Simply Supported-Simply Supported (SS-SS). Analytical solution is carried out using Euler-Bernoulli beam theory and Newton Raphson Method. First, the equations of motion are provided. Then, solutions including the effects of the geometric characteristics, and boundary conditions are obtained and discussed for the natural frequencies of the first three modes. To confirm the reliability of the vibration analysis carried out in the present paper as well, all the analytical results are checked with the corresponding numerical results obtained from the finite-element-method (FEM) based software called ANSYS. Numerical and analytical results are found to be good agreement.

Nonlocal nonlinear free vibration of functionally graded nanobeams

In this paper, nonlinear free vibration of functionally graded (FG) nanobeams with immovable ends, i.e. simply supported-simply supported (SS) and simply supported-clamped (SC), is studied using the nonlocal elasticity within the frame work of Euler–Bernoulli beam theory with von kármán type nonlinearity. The material properties are assumed to change continuously through the thickness of the FG nanobeam according to a power-law distribution. The analytical solution for the nonlinear natural frequency is established using the method of multiple scale. The small scale effects on the linear/nonlinear nonlocal frequency to the linear/nonlinear classical frequency ratios (the linear/nonlinear frequency ratios) are examined for various parameters such as the FG nanobeam length, the FG nanobeam thickness to length ratio (the thickness ratio), the vibration amplitude to the radius of gyration ratio (the amplitude ratio), and the boundary condition. As a main result, it is observed that while the linear frequency ratios are independent of the gradient index, the nonlinear frequency ratios vary with the gradient index.

Analysis of Forced Vibrations in Micro-Scale Anisotropic Thermo-Elastic Beams Due to Concentrated Loads

This article is devoted to model and analyze the transverse deflection and thermal moment of transverse vibrations in a transversely isotropic, thermo-elastic beam resonator under the action of time harmonic concentrated load. The governing equations of flexural vibrations and thermal moment in a transversely isotropic, thermo-elastic Euler–Bernoulli beam have been derived in a closed form. A time harmonic point load is assumed to act on the beam at a given distance from the origin. The beam is assumed to be at either clamped-clamped (CC), simply supported-simply supported (SS), clamped-simply supported (CS), or clamped-free (CF) conditions at its ends. The Laplace transform technique has been used to find the transverse deflection and thermal moment in the transform domain due to the action of concentrated load in a beam under above conditions in case of both free and forced vibrations. The analytic expressions obtained in the physical domain after inversion of Laplace transforms have been computed numerically with the help of MATLAB software for silicon material beam. The results for coupled thermo-elastic, elastic, and isotropic beams have been deduced as special cases. The computer-simulated results have been presented graphically. The study may find applications in development and design of resonators (sensors), precision thermometers, and energy harvesters.

Thermoelastic vibration analysis of Mems/Nems plate resonators with voids

In this paper, the effect of voids, relaxation times, thermomechanical coupling, surface conditions, and plate dimensions on energy dissipation induced by thermoelastic damping in microelectronics mechanical systems (MEMS)/ nanoelectronics mechanical systems (NEMS) resonators are investigated. Closed form expressions for the transverse vibrations of a homogenous isotropic, thermoelastic thin plate with voids, based on Kirchhoff theory have been derived. The exact solutions for the free vibrations of plates under clamped-simply supported (CS) and simply supported-simply supported (SS) conditions are obtained. Analytical expressions for deflection, temperature change, frequency shifts, and thermoelastic damping in the plate have been derived. Some numerical results with the help of MATLAB programming software in case of silicon nitride and magnesium like material have also been presented.

Vibration analysis of cross-ply laminated composite cylindrical shells using the wave propagation approach

The wave propagation approach is extended to analyse the natural frequencies of cross-ply laminated composite cylindrical shells. Results obtained using the method have been evaluated against those available in the literature and good agreement has been found. With the present method the effects of shell parameters on the frequencies are investigated. The boundary conditions considered are clamped/clamped (CC), clamped/simply supported (CSS), simply supported/simply supported (SSSS) and clamped/sliding (CSL) conditions. The influences of boundary conditions can be seen to be more significant at small circumferential mode n than at large circumferential modes. There are transitions of fundamental frequencies from the n=3 curve to the n=2 curve with the h/R ratios. The differences of frequency between different E 11 /E 22 ratios are significant at low circumferential modes but not so significant at large circumferential modes.

Deformation of varying thickness composite cylindrical shell subjected to fluid loading with various end conditions

A horizontal orthotropic multi-layered cylindrical vessel subjected to fluid loading is analyzed by finite element displacement method. The end conditions considered are simply supported-simply supported (SS), clamped-clamped (CC), clamped-free (CF). In the case of varying thickness shell the thickness is varied piecewise linearly. Numerical results are presented for various cases in the form of graphs and tables.