Non-dimensional Frequency Parameters Research Articles

Vibration analysis of sandwich functionally graded material plate with cut-outs using artificial neural network technique

In the present article, the effect of geometric discontinuities on the vibrational response of porous sandwich functionally graded material (SFGM) plates with double FGM facesheets has been investigated using the artificial neural network (ANN) technique. Generalized governing equations for the SFGM plate have been derived based on nonpolynomial based higher-order shear deformation theory (HSDT). Geometric discontinuities have been incorporated in terms of cut-outs in the SFGM plates. The FGM layers integrate a porosity model, while the core layer is considered a ceramic layer within the SFGM plate structure. Further, a C° continuous isoparametric finite element formulation with a four-noded, isoparametric quadrilateral element with seven degrees of freedom (DOFs) per node has been employed to accomplish the results. The accuracy of the present results has been demonstrated through convergence and validation studies. A comprehensive study has been carried out to investigate the influence of volume fraction index, even and uneven porosity distribution and cut-outs on the frequency parameter of SFGM plates. However, the finite element method (FEM) is computationally challenging. Therefore, the motivation behind adopting ANN technique to develop a predictive model for reasonable accuracy with less computational time. The ANN technique is proposed to predict the NDFP of the SFGM plate using cut-outs under various conditions using numerical simulation datasets. An optimised ANN model has been developed with an architecture of (6–5–10–5–1), exhibits best performance based on mean error value, which accurately predicts the Non-Dimensional Frequency Parameter (NDFP) of the SFGM plate.

Computer model of non-Newtonian canalicular fluid flow in lacunar-canalicular system of bone tissue.

Brittle bone diseases are a global healthcare problem for orthopaedic clinicians, that reduces bone strength and promotes bone fracture risk. In vivo studies reported that loading-induced fluid flow through the lacunar-canalicular channel (LCS) of bone tissue inhibit such bone loss and encourages osteogenesis i.e. new bone formation. Canalicular fluid flow converts mechanical signals into biological signals and regulates bone reconstruction by releasing signalling molecules responsible for mechanotransduction. In-silico model mostly considers canalicular fluid is Newtonian, however, physiological canalicular fluid may be non-Newtonian in nature as it contains nutrients and supplements. Accordingly, this study attempts to develop a two-dimensional in-silico model to compute loading-induced non-Newtonian canalicular fluid flow in a complex LCS of bone tissue. Moreover, canalicular fluid is considered as a Jeffery fluid, that can easily be reduced to Newtonian fluid as a special case. The results show that physiological loading modulates the canalicular fluid flow, wall shear stress (WSS) and streamline in bone LCS. Fluid velocity and WSS increases with increase in non-dimensional frequency and non-Newtonian parameter (Jeffery fluid parameters) and reduce with change in permeability. The outcomes of this study may provide new insights in the role of mechanical loading-induced non-Newtonian canalicular fluid flow dynamics in bone LCS. The key findings of this study can be used to improve the understanding of osteocyte mechanobiology involved inside the bone tissue.

Transverse free vibration analysis of thin sectorial plates by the weak form quadrature element method

The weak form quadrature element method is applied to free vibration analysis of thin sectorial plates with arbitrary vertex angles and boundary conditions. To tackle the strong stress singularity around the vertex, analytical displacement descriptions are introduced into the inner sectorial subdomain, while the outer annular subdomain is modeled by a single weak form quadrature thin plate element. The continuity on the interface between the two subdomains is enforced afterwards. Eventually, a generalized eigenvalue formulation is established after introducing Hamilton’s principle. The first six non-dimensional frequency parameters for various vertex angles and boundary conditions are obtained and compared with available results. Several typical free vibration modes are plotted. The accuracy, convergence rate, and computational cost of the present formulation are discussed at length.

Free vibration analysis of FGM spherical and elliptical shells under nonlinear thermal environments

The thermo-mechanical effects of the free vibration of functionally graded (FG) spherical and elliptical shells under nonlinear temperature distributions (NTD) are investigated. The geometries of the FG spherical and elliptical shells are expressed in terms of the surface fundamental form, and the displacement function is derived based on higher-order shear deformation theory (HSDT). Temperature-dependent (TD) material and heat conduction are investigated based on Voigt's micromechanical model. The equation of motion of the thermo-elastic FG shells is formulated using the classical Hamilton's principle. Numerical results are obtained using the finite element method with quadrilateral Lagrangian isoparametric elements. A comparison of the results for the free vibration of FG shells based on the FSDT and HSDT approaches is presented. The first fundamental mode is the axisymmetric mode for the circular plate and oblate shell, but the results are otherwise for spherical and prolate shells. The torsional mode occurs in the seventh mode within the first ten vibrational modes for the prolate shell. In addition, the frequency parameter in the torsional mode is independent of the different support conditions. Parametric studies show that the non-dimensional frequency parameters of the FG shells decrease quickly, then slowly, as the volume fraction index increases. Finally, the frequency parameters decrease when the temperature at the top of the FG shell surface increases.

Significant influence of fluid viscoelasticity on flow dynamics past an oscillating cylinder

This study focuses on numerically investigating the impact of fluid viscoelasticity on the flow dynamics around a transversely forced oscillating cylinder operating in the laminar vortex shedding regime at a fixed Reynolds number of $Re = 100$ . Specifically, we explore how fluid viscoelasticity affects the boundary between the lock-in and no lock-in regions and the corresponding wake characteristics compared with a simple Newtonian fluid. Our findings reveal that fluid viscoelasticity enables the synchronization of the vortex street with the cylinder motion at lower oscillation frequencies than those required for a Newtonian fluid. Consequently, the lock-in region boundary for a viscoelastic fluid differs from that of a Newtonian fluid and expands in the non-dimensional cylinder oscillation amplitude and frequency parameter space. In the primary synchronization region, the wake of a Newtonian fluid exhibits ‘2S’ (two single vortices) and ‘P+S’ (a pair of vortices and a single vortex) shedding modes. In contrast, a ‘2P’ (two pairs of vortices) vortex mode is observed for a viscoelastic fluid within the same region. To gain a deeper understanding of the differences in the coherent flow structures and their associated frequencies between the two fluids, we employ the data-driven reduced-order modelling technique, known as the dynamic mode decomposition (DMD) technique. Utilizing this technique, we successfully extract and visualize the two competing fundamental frequencies (cylinder oscillation and natural vortex shedding frequencies) and their associated flow structures in the case of the no lock-in state, whereas only the dominant cylinder oscillation frequency and associated flow structure in the case of the lock-in state. Furthermore, we propose that the presence of excess strain resulting from the stretching of polymer molecules in viscoelastic fluids leads to a distinct difference in the wake structure compared with Newtonian fluids. This observation aligns with the findings obtained from the $Q$ -criterion and vorticity transport analysis of the wake.

Free vibration analysis of thin plates with side cracks by the weak form quadrature element method

The weak form quadrature element method is applied to vibration analysis of thin plates with side cracks. Eigenfunctions are introduced into the inner square domain around the crack tip, while the rest of the problem domain is analyzed by the weak form quadrature element method (QEM). The continuity of displacements and slopes on the interface between the inner and outer domains is enforced. A generalized eigenvalue formulation is established after introducing Hamilton’s principle. The first ten nondimensional frequency parameters for several typical problems are obtained and compared with those available in the literature. The corresponding modes of free vibration are plotted, and the influence of the crack on vibration frequencies and modes is discussed at length.

Natural Frequencies of a Viscoelastic Supported Axially Functionally Graded Bar With Tip Masses

The frequency parameters of an Axially Functionally Graded (AFG) bar for viscoelastic and point mass boundary conditions are studied in this paper. The structures with functionally graded materials under axial loads can be modelled as AFG bars. Those may be considered as support elements with spring and dashpot in the vibration isolation. Keeping the ratio of force frequency and natural frequency at a certain level is ensured to control the natural frequency therefore dynamic amplification factor with different material combinations. Various boundary conditions are attained by changing the spring and damping coefficients of viscoelastic support elements and the ratio of rod mass to tip mass. Researchers assume that the beam material properties in directions of length and thickness change exponentially, individually, or both in their studies. There are a few studies on the AFG rods in the existing studies. Analysis is carried out via the finite element method for the non-dimensional frequency parameters of the bar in MATLAB. The energy equations of the motion are obtained in the frame of axial bar theory considering the material properties of the bar vary longitudinally according to the power-law distribution. The effects of material distribution, spring, damping and tip mass values on the bar's frequency parameters and structural behaviour have been extensively investigated.

Novel size-dependent finite element formulation for modal analysis of cracked nanorods

It is a remarkable engineering problem how possible structural damages in nano/micro scaled mechanical elements affect the working performance and mechanical behavior of nano/micro-electro-mechanical system (NEMS/MEMS) organizations. There are various nano/micro-scale mechanical elements that can be modelled as one-dimensional and continuous systems in NEMS/MEMS organizations. With this motivation, the free vibration analysis of nano scaled axial rods with a single crack is examined by employing the nonlocal finite element formulation for the first time in the literature in this study. Firstly, the differential equation of nonlocal vibration of nanorods is presented for a single crack. Next, a nonlocal finite element formulation developed for the case of crack is proposed. The local and nonlocal nondimensional frequency parameters of cracked nanorods are compared with some studies in the literature. Additionally, nondimensional frequencies of nanorods are presented with several tables and graphics according to different parameters such as nonlocal parameter, mode number, crack intensity and crack location. Detailed discussions of the numerical results are given, and the conclusions deduced from these discussions are summarized.

FOLDED ANGLE DEPENDENT MODAL ANALYSIS OF THE FLAT AND FOLDED PLATES

The effect of folded angle and boundary condition differences on the natural frequency, which is one of the dynamic characteristics of isotropic thin plates, is investigated. Modal analyzes of flat and folded structures are performed with ANSYS under cantilever and two-side-fixed boundary conditions. The validation of the free vibration results of the structures is performed with Solidworks. The first five natural frequencies of each structure are obtained, and the values in terms of non-dimensional natural frequency parameters are interpreted with the tables and graphics. While both an increase and a decrease can be observed depending on the increase in the folded angle in the first five natural frequencies for the cantilever boundary condition, the effect of the increase in the folded angle on the natural frequencies in the two-sided-fixed structures is much less in the higher folded angle values. It is seen that the reason for the sudden changes in the folded angle - non-dimensional natural frequency graphs is the mode shape change.

Temporal Spectrum of a Scattered Electromagnetic Waves in the Conductive Collision Turbulent Magnetized Plasma

Using WKB method the peculiarities of the temporal spectrum of an ordinary and extraordinary electromagnetic wave scattered in a weakly randomly inhomogeneous three-dimensional nonstationary and conductive magnetized plasma are investigated. On the basis of the stochastic differential transport equation for the frequency fluctuation the broadening and the displacement of the temporal spectrum for both waves are obtained for the polar terrestrial ionosphere. These statistical characteristics contains anisotropic parameters: velocity of a plasma stream, conductivities of the ionosphere, elongated electron density irregularities are characterized by the anisotropy factor and inclination angle with respect to the geomagnetic lines of forces. The analysis of the power spectrum of the waves as a function of the propagation distance and the nondimensional frequency parameter containing the carrier frequency and characteristic temporal scale of electron density fluctuations are carried out. Analytical and numerical calculations have shown that the terrestrial conductivity and anisotropy factors exert a substantial influence on evaluation of the temporal spectrum than the inclination angle. It was found that the wave spectrum increases initially as the square root of the propagation distance, but at large distances it approaches a limiting value. Statistical moments of a scattered ordinary and extraordinary waves do not depend on an absorption sign and valid for both absorptive and active media.

Compressor-Surge Behaviors Distinguished Between Near-resonant Type and Convective Type

In deep surges in compressors, the behaviors tend to show either a near-resonant one or a convective one, the latter performing emptying and filling actions. However, detailed information on the differences between them have not been obtained so far. In order to understand the situations, surge simulations by a one-dimensional simulation code were conducted with respect to a single-stage centrifugal compressor, paying attention to the effects of changes in the compressor tip speed and in the geometry of the flowpaths. As the results, it has been found that near-resonant surges appear at very low tip speeds and for relatively narrow sectional areas of the delivery flowpath, and they transform to convective surges of emptying and filling actions toward higher speeds and wider sectional areas. The features of the behaviors are generalized in terms of several non-dimensional surge frequency parameters. The results are visualized by several global framework maps showing the relations among the parameters. The obtained results will give deeper understandings on the surge phenomena.

Modal analysis of E-glass basalt laminated composite plates based on experimental mechanical properties using FEA

ABSTRACT The research work presents the comparative study on finite element modal analysis of 14 layers E-glass epoxy (G/E plate) and 10 layers basalt epoxy (B/E plate) composite angle-ply trapezoidal plate for various boundary conditions. The G/E and B/E plate having fiber orientations of [0°/+45°/0°/-45°/0°/+45°/0°]s and [0°/+45°/0°/-45°/0°]s are considered for fabricating plates using compression molding technique to minimize void content. Then, orthotropic properties are determined by experimental testing as per ASTM standards. The effects of different geometrical parameters including aspect ratio (a/b = 1 and 2), taper ratio (c/b = 0.25–1), and span-to-thickness ratio (a/h = 50) of trapezoidal plates on undamped modal analysis are done for G/E and B/E plates. A shell 281 element, which obeys first-order shear deformation theory, is applied in commercial finite element software to extract the natural frequencies and mode shapes for various boundary conditions using properties determined by experimental testing. The calculated mechanical properties are compared between G/E plate and B/E plate, and the results show basalt fiber as alternative material to E-glass fiber. The results of nondimensional frequency parameters and mode shapes of G/E and B/E plate also show good agreement with previous research work.

Calculating the fundamental frequency of power law functionally graded beam using ANSYS software

In this work free vibration of simply supported, clamped-clamped and clamped-free functionally graded (FG) beam with material graduation transversally through the thickness, using the power-law model, were investigated. The beam’s functionally graded material (FGM) consists of aluminium (Al2O3) and steel (used in a typical case). Pure steel is the beam’s bottom surface, whereas pure aluminium is the beam’s top surface throughout the thickness change condition. Two finite element models were proposing to calculate the first non-dimensional frequency parameters of FG beam. These models are shell and solid, and they were employed using the ANSYS APDL version 17.2. The two models have been verifying with the previously published works, and a good agreement was founding. Numerical results were presented in graphical forms to study the effects of the power-law index (i.e. material distribution), length-to-thickness ratio, modulus ratio and types of support for the first non-dimensional frequency parameter of the FG beam. The above mention effects play a significant role in the free vibration of the beam. Power-law index (k) is one of the parameters affecting mainly on the frequency parameter of FG beam. The frequency parameter increase with increasing the power-law index when the modulus ratio less than one and decease with increasing the power-law index when the modulus ratio more than one, when the modulus ratio equal one ( i.e. pure material ), there is no effect of the power-law index in this case. When the modulus ratio (E ratio) increases, the frequency parameter increases too, but with the change rate depending on the length-to-thickness ratio (L/h) and power-law index(K). Also, the frequency increase with increasing the length to thickness ratio at any power-law index.

Vibration and dynamic behavior of electrostatic size-dependent micro-plates

The recent trend of using thin micro-plate structures in severe operational conditions caused the classical theory (CT) to be no longer suited in analyzing the dynamic characteristics of them. For this study, the modified version of the couple stress theory (MCST) is adopted. Then, using Hamilton’s principle and Kirchhoff plate theory, the dynamic form of the size-dependent equation of motion for micro-plates stimulated by electrostatic dynamic excitation is acquired. The mixed extended Kantorovich and differential transformation methods on partial differential equations are applied to obtain the mode shapes and natural frequencies. The mode shapes from the linear free vibration solution are used as the mode shapes of the assumed multi-mode displacement field. The displacement field is required to solve the dynamical equation of motion. The equations are subsequently solved by the fourth-order Runge–Kutta method. Some universal graphs are also presented using the MCST to predict the effects of the size effect, tensile and compressive residual stresses, and aspect ratio of micro-plate on the free vibration and dynamic behaviors of micro-plates around their static configuration. It is found that the non-dimensional natural frequency parameter varies linearly versus the other non-dimensional parameters of the micro-plate. Various interesting phenomena are observed from the current simulations of the dynamic response of micro-plates to both DC and AC excitation, as well as the boundary of the frequency behavior conversion. A comparison is drawn between the responses of the micro-plate based on the modified couple stress and CT taking the size dependency into account. To facilitate understanding of the physical phenomena observed in the results, a simple physical analog to the micro-plates is suggested. This physical analog, which describes the boundary of frequency behavior conversion, is the linear relationship between the non-dimensional parameters of the system. The study of this model can help to realize some of the complex responses of the micro-plates.

Three-Dimensional Vibration Analysis of Laminated Composite Rectangular Plate with Cutouts.

The main purpose of this paper is to establish an analysis model of laminated composite rectangular plate with/without cutouts on the basis of three-dimensional elasticity theory and provide the exact three-dimensional solution. In the present work, the effect of the cutout is considered by subtracting the energies of cutouts from the total energy of the entire plate. The standard three-dimensional trigonometric cosine Fourier series and auxiliary Fourier series are chosen as admissible functions, and the Hamilton’s principle and Rayleigh-Ritz procedure are used to obtain the exact solution. In order to verify the effectiveness and reliability of the proposed method, some numerical results are obtained, and the results are compared with the ones available in the literature or finite element analysis. Finally, the effects of some key parameters which will affect the vibration characteristics are analyzed, and the non-dimensional frequency parameters are obtained, which can serve as benchmark data for the future research.

Nonlocal finite element analysis for axial vibration of embedded love–bishop nanorods

In this paper, nonlocal free vibration of axial rods embedded in elastic medium is examined by utilizing Love–Bishop rod theory with finite element method for the first time in the literature. Firstly, kinematic relations and dynamic equilibrium of this rod formulation are constituted. Equation of motion and boundary conditions are attained with the help of first variation of total potential of nano rod and solved using separation of variables. The frequency equations of four different nanorod types are obtained. On the other hand, a size–dependent finite element formulation is presented based on Weighted Residual Method. The nondimensional frequency parameters of nanorods are calculated by using analytical and finite element methods for different parameters such as mode number, nonlocal parameter, rod length, length–to–diameter ratio, finite element number and a detailed discussion is performed in the numerical results. In order to understand the characteristics of this rod theory, several comparative results that include nonlocal simple rod formulation are given as well.

Size-dependent vibration behavior of graphene sheet with attached spring-mass and damper system based on the nonlocal Eringen theory

The aims and objectives of the current study are to investigate the vibration analysis of the graphene sheet with attached spring, mass, and damper system for the first time. For this purpose, the governing equation is derived employing Hamilton’s principle regarding the nonlocal Eringen theory. The graphene sheet is supposed to be rested on a Winkler elastic foundation. Galerkin’s method is being used to obtain frequency on various boundary conditions. To validate our work, the proposed model is demonstrated by comprising the present model with the results available in the literature. The influence of different parameters such as the attached mass, nonlocal parameter, aspect ratio, and Winkler foundation of a graphene sheet on the non-dimensional vibration frequencies are presented in detail. The results indicate that adding the spring-mass system leads to an increase in the non-dimensional frequency parameter. The outcomes of this survey can be utilized in the construction of biological and chemical nanosensors.

Hygro-Magnetic Vibration of the Single-Walled Carbon Nanotube with Nonlinear Temperature Distribution Based on a Modified Beam Theory and Nonlocal Strain Gradient Model

In this study, vibration analysis of single-walled carbon nanotube (SWCNT) has been carried out by using a refined beam theory, namely one variable shear deformation beam theory. This approach has one variable lesser than a contractual shear deformation theory such as first-order shear deformation theory (FSDT) and acts like classical beam approach but with considering shear deformations. The SWCNT has been placed in an axial or longitudinal magnetic field which is also exposed to both the hygroscopic as well as thermal environments. The thermal environment is considered as nonlinear thermal stress field based on the Murnaghan’s model whereas the hygroscopic environment is assumed as a linear stress field. The size effect of the SWCNT has been captured by both the nonlocal and gradient parameters by employing the Nonlocal Strain Gradient Theory (NSGT). Governing equation of motion of the proposed model has been developed by utilizing the extended Hamilton’s principle and the non-dimensional frequency parameters have been computed by incorporating the Navier’s approach for Hinged–Hinged (HH) boundary condition. The proposed model is validated with the existing model in special cases, by comparing the non-dimensional frequency parameters, displaying an excellent agreement. Further, a parametric study has been conducted to analyze the impact of nonlocal parameter, gradient parameter, thermal environment, hygroscopic environment, and magnetic field intensity on the non-dimensional frequency parameters. Also, results for some other theories like Classical Elasticity Theory (CET), Nonlocal Elasticity Theory (NET), and Strain Gradient Theory (SGT) have been presented along with the NSGT.

Free vibration of FGM annular sectorial plates with arbitrary boundary supports and large sector angles

An original work is presented for the free vibration analysis of thin functionally graded material (FGM) annular sectorial plates with arbitrary boundary supports using the weak form quadrature element method (QEM). Novel C 1 quadrature annular sectorial plate element is developed based on the physically neutral surface thin plate theory in cylindrical coordinate system. Three displacement functions are used to avoid the introduction of the mixed second order derivative degree of freedom at corner points. The derived explicit formulations with a varying node number can be applied for any type of node. A way to define the elastic spring constants is proposed to ensure the independent of the non-dimensional frequency parameter on the power-law exponent. The convergence and accuracy of the present model are validated by comparing the obtained results with those available in literatures. It is demonstrated that the proposed model is simple, straightforward and has good performance. A wide range of numerical results are reported to demonstrate the effect of the sector angle, ratio of inner/outer radii, and boundary conditions on the vibration behavior of the thin annular sectorial plate. Some tabulated results should be new which may serve as a reference for verifying numerical solutions obtained by other methods.

Dispersion equation of Rayleigh waves in transversely isotropic nonlocal piezoelastic solids half-space

This study is devoted to investigate the propagation of Rayleigh-type waves in transversely isotropic nonlocal piezoelastic half-space. When the stress-free boundary is maintained at charge-free condition, the dispersion equation for the propagation of Rayleigh waves at the free surface of transversely isotropic piezoelastic solids has been obtained. Based on the obtained dispersion equation, the effect of the nonlocality on the speed of Rayleigh wave is numerically considered. The dependence of velocities of plane waves in transversely isotropic nonlocal piezoelastic medium on the direction of propagation as well as non-dimensional frequency parameter has been also illustrated.