Non-dimensional Natural Frequency Research Articles

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.

Computational modeling of sigmoid functionally graded material (SFGM) plate

The first-order shear deformation theory (FSDT) was used to explore the natural frequency response of functionally graded piezoelectric plates subjected to static electrical and mechanical strain in this present study. A monomorph model for a functionally graded piezoelectric plate with material properties that change according to sigmoid law with respect to plate thickness has been considered. A three-dimensional finite element model with a free tetrahedral element mesh was created using COMSOL 4.2 Multiphysics® software, with each node having three degrees of freedom. Variations in the FGPM plate’s free vibration boundary conditions, composition, and geometry have all been investigated. In free vibration analysis, non-dimensional natural frequency of FGPM plate initially decreasing considerably and then remaining almost constant with the increase in volume fraction index when material property graded by power law. When material properties are varied by sigmoid law, with an increase in volume fraction index, the non-dimensional natural frequency of FGPM plates remains virtually constant. FGPM plates have a lower non-dimensional natural frequency if the thickness to width ratio is greater. Non-dimensional natural frequencies of Clamped-Clamped FGPM plates (C–C–C–C) are greater compared to Clamped-Free FGPM (C–F–C–F) and Simply Supported Free FGPM (S–F–S–F).

Simultaneous streamwise and cross-stream oscillations of a diamond oscillator at low Reynolds numbers

A two-dimensional stabilized space-time finite-element-based in-house solver is used to explore flow around an elastically mounted rigid diamond oscillator undergoing undamped vortex-induced vibrations in transverse and stream-wise directions simultaneously. The computations are carried over a reduced velocity (U*) range of 1–12 keeping the Reynolds number (Re) and the mass ratio (m*) fixed at 100 and 10, respectively. In the current investigation, a five-branch response behavior with the presence of an “upper branch” is identified, which is absent for transverse-only oscillations. A shift in normalized time-averaged phase difference (ϕ/π) from 0 to 1 is related to the transition from the upper to the lower branch. The non-dimensional oscillation frequencies in both streamwise and cross-stream directions (Fx and Fy, respectively) collapse on the upper branch, matching the oscillator's non-dimensional natural frequency (FN). This leads to periodic single-looped “raindrop-shaped” cylinder trajectories. On this branch, the vortex-shedding mode is asymmetric, and the mean lift is non-zero (Clavg≠0). For all other response branches, Clavg≈0 and the vortex-shedding modes are symmetric. The presence of multiple frequencies in the in-line oscillations makes the initial branch quasi-periodic, whereas all other response branches are periodic. The addition of in-line motion modifies the fluid loading on the oscillator significantly.

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.

Natural frequency analysis of imperfect FG sandwich plates resting on Winkler-Pasternak foundation

The present research analyzes the natural frequencies of the imperfect functionally graded sandwich plate (FGSP) comprised of porous face sheets made of functionally graded materials (FGMs) and an isotropic homogeneous core resting on the elastic foundation. To accomplish this, the material characteristics of the FGSP are taken to be changed incessantly along the thickness direction based on the volume fraction of constituents expressed by the modified rule of the mixture, which includes porosity volume fraction with three diverse kinds of porosity distribution models. Furthermore, to describe the two-parameter elastic foundation's response on the imperfect FGSP, the medium is supposed to be linear, homogenous, and isotropic, and it has been modelled using the Winkler-Pasternak model. Moreover, in the kinematic relationship of the imperfect FGSP resting on the Winkler-Pasternak foundation, third-order shear deformation theory (TSDT) is used, and the motion equations are set up employing Hamilton's principle. For natural frequency analysis of imperfect FGSPs resting on the Winkler-Pasternak foundation with simply supported edges, an analytical solution is obtained. To verify the current formulation, comprehensive comparisons are performed with the available data. The impacts of the two-parameter elastic foundation, porosity volume fraction, porosity distribution types, lay-up scheme, and side to thickness ratio, on the values of the non-dimensional fundamental natural frequency (NDFNF) of the imperfect FGSPs, are investigated, thoroughly.

Buckling and Prestressed Vibrations of Periodic Nonprismatic Beams Using Integral Equation Approach

This paper presents the use of Fredholm–Volterra integro-differential equation approach for determining the critical buckling load and natural frequencies of periodically supported nonprismatic Euler–Bernoulli beams. Neumann series solution is used to solve the integro-differential equation by using the orthogonal Chebyshev polynomials, to avoid ill-conditioning. Gauss quadrature is used to perform various needed integrals. Critical buckling load, natural frequencies, and the corresponding mode shapes are compared with those available in the literature, those obtained using the principle of virtual work, and the ones obtained using NASTRAN®. The effect of variation of taper ratio, both in width and height of a rectangular cross section, on the nondimensional critical buckling load and the nondimensional natural frequency, is also investigated. A singularity is observed near the taper ratio of unity, which impacts the accuracy of both the finite element and the principle of virtual work results. The integral equation approach converges with a lesser number of terms than the principle of virtual work, especially for higher modes for beams approaching the taper ratio of unity. Benchmark examples for free vibration of a multiple-bay beam and of inflated tube are also studied, and the results are compared with those presented in the literature.

Applying nonlocal strain gradient theory to size-dependent analysis of functionally graded carbon nanotube-reinforced composite nanoplates

In this research paper, as initial endeavors, the vibrational responses of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) nanoplates taking into account the effect of nonlocal parameter and strain gradient coefficient are investigated. The study aims at developing mathematical modeling via an analytical solution to FG-CNTRC nanoplate structure with allowance for the nonlocal strain gradient effect. The four types of CNT distribution are used and compared in the context of the vibration of nanoplate in the presence of the small length scale effects, namely the (a) UD, (b) FG-V, (c) FG-O, and (d) FG-X. Some theoretical equations based on the first-order shear deformation plate theory (FSDT) are presented to provide a lucid understanding of how the small length-scale influences the FG-CNTRC nanoplate. For the vibrational analysis of a nanoplate, which is simply supported boundary condition, Navier solutions are obtained. Also, in contrast to earlier studies, an analytical approach is used to establish the governing equations of the FG-CNTRC nanoplate. Some specific numerical examples are given and compared with the results presented in the literature. In the section of numerical results, the influence of the nonlocal parameter, strain gradient coefficient, geometric parameters and vibrational modes on the non-dimensional natural frequency are investigated and discussed in detail. These could be useful to analysts and designers to estimate the fundamental natural frequencies in each of the four CNT distributions that the FG-CNTRC nanoplate possesses.

Response analysis of hybrid functionally graded material plate subjected to thermo-electro-mechanical loading

Static and free vibration response analysis of a functionally graded piezoelectric material plate under thermal, electric, and mechanical loads is done in this study. The displacement field is acquired using the first-order shear deformation theory, and the Hamilton principle is applied to deduce the motion equations. Temperature-dependent material properties of the functionally graded material plate are used, and these properties follow the power-law distributions along the thickness direction. However, the properties of piezoelectric material layers are assumed to be independent of the electric field and temperature. Finite element formulation for the functionally graded piezoelectric material plate is done using the combined effect of mechanical and electrical loads. The effects of parameters like electrical loading, volume fraction exponent N, and temperature distribution on the static and free vibration characteristics of the functionally graded piezoelectric material square plate are analyzed and presented. Responses are obtained in terms of the centerline deflection, axial stress and the nondimensional natural frequency with various boundary conditions. It is observed that the centerline deflection and nondimensional natural frequency increases as exponent N increases. At the same time, the axial stress decreases with an increase in exponent N. The findings of the static and the free vibration analysis suggest the potential application of the functionally graded piezoelectric material plate in the piezoelectric actuator as well as for sensing deflection in bimorph.

Closed form solution for dynamic analysis of rectangular nanorod based on nonlocal strain gradient

ABSTRACT In this study, the free torsional vibration of the nanorod with noncircular cross-section based on the nonlocal strain gradient theory is investigated. The rectangular cross-section has been selected. The governing partial equation of the torsional vibration of the noncircular nanorod is derived using Hamilton’s principle. Using a Galerkin method, and by consideration of the rectangular shape function in series, the natural frequency of noncircular nanorod is obtained. The obtained results indicate the convergence of the nondimensional natural frequency of the nanorod by enhancement of the number of the rectangular shape function. Furthermore, the nondimensional natural frequency of the noncircular nanorod for different nondimensional nonlocal parameter and nondimensional material length scale parameter is studied. Eventually, the effects of both nonlocal parameter and material length scale parameter on the nondimensional natural frequency, which causes torsional stiffness softening and torsional stiffness hardening of the nanorod, is also illustrated.

Influence of non-circular cross section shapes on torsional vibration of a micro-rod based on modified couple stress theory

Torsional micro-rods are key components in many micro-sensors and micro-resonators in aviation industry but in the literature proposed models have usually circular cross section which can be a limitation. In the torsional vibrations of micro-structures with non-circular cross section shape, the displacement fields and governing equations are completely dependent on the cross section shape of the structure which makes the analysis procedure more complicated. The aim of this study is to demonstrate the impact of geometrical parameters of the micro-rod especially its cross section shape, the type of medium and the presence of the external fields such as magnetic field in the torsional vibration behavior of the micro-rod. For this purpose, using the Hamilton's principle, the governing equation of the torsional vibration of the micro-rod with non-circular cross section embedded on elastic medium in the presence of a magnetic field is derived. To show the effects of size dependency of the micro-rod, modified couple stress theory is used. For instance, elliptical and rectangular cross section shapes with the same area are chosen for a micro-rod. After calculating the natural frequency of the micro-rod by the Galerkin method, the effect of cross section shape on the non-dimensional natural frequency of the micro-rod is investigated for changes in the aspect ratio of the cross section. In addition, the influence of length scale parameter, stiffness of torsional medium and magnetic field permeability on the natural frequency of the micro-rod are evaluated. The obtained results demonstrate that the natural frequency of the micro-rod is completely affected by the shape and aspect ratio of the cross section. These results can be useful in the micro-structure design stage. Also, a comparison is made between the obtained analytical results and those of semi-analytical differential transform method (DTM).

Free vibration analysis of hybrid laminated plates containing multilayer functionally graded carbon nanotube-reinforced composite plies using a layer-wise formulation

This paper studies the free vibration behavior of hybrid laminated plates using a layer-wise formulation. The laminated structures are made of graphite fiber-reinforced composite (GFRC) and multilayer functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plies. The internal single-walled carbon nanotubes (SWCNTs) are distributed in layer-wise form along the ply thickness direction either uniformly or functionally graded according to four separate patterns. Both the linear and nonlinear of the CNTs distributions form are considered. Theatrical formulations are based on the first-order shear deformation theory (FSDT), and finite element method is employed to obtain the nondimensional natural frequency. The elastic modulus of both GFRC and CNTRC plies composed the hybrid laminated plate is obtained by a micromechanics model. Initially, the provided results are validated by comparing it with the literature; thereafter, a parametric study is carried out the influences of laminates configuration, number of CNTRC plies layers, CNT volume fraction, CNT distribution patterns, linear and nonlinear distribution of CNTs, plate width-to-thickness ratio, plate aspect ratio, boundary conditions, stacking sequences, and number of plies on the free vibration behavior of hybrid laminated plates. Some final remarks considering the laminated plate configuration and the distribution form of the CNT fillers are presented in order to provide a useful observation on the design criteria of the laminated composite plate structures.

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.

Nonlocal Strain Gradient Pull-in Study of Nanobeams Considering Various Boundary Conditions

The main objective of this investigation is to study the size-dependent dynamic pull-in instability of nanobeams based on the nonlocal strain gradient theory (NLSGT) and Euler–Bernoulli beam model. To this end, the partial differential equation is obtained based on the NLSGT considering the electrostatic, fringing field, and intermolecular nonlinear forces. Then, the Galerkin method and the homotopy analysis method (HAM) were employed to solve the nonlinear governing equation. To validate the proposed results, the non-dimensional natural frequency and pull-in voltage are compared with the previously published results. Likewise, the analytical results of the HAM are compared with those obtained based on the Runge–Kutta numerical method. Besides, the impacts of the NLSGT, strain gradient theory, nonlocal theory, and classical theory on the dynamic behavior of nanobeams are investigated in the same situation. The pull-in voltage is also presented and the effects of electrostatic forces, fringing field, and initial gap are discussed in detail for different boundary conditions.

Non dimensional analysis of axially polarized passive magnetic bearings

Ranges of investigation on passive magnetic bearing exist where radial and axial forces and variations of a few coefficients of radial and axial stiffnesses are shown for a particular set of chosen parameters and dimensions. The paper presents a non-dimensional approach for the same with additional parameters like all coefficients of stiffnesses and natural frequency. The model is considered for static analysis of axially polarised ring shaped magnets for both stator and rotor part in three degrees of freedom with three linear translations in 3D Cartesian coordinate system. The significance of this non-dimensional method is more generalised to the designer where boundary parameters like maximum force on the rotor of passive magnetic bearing, natural frequency, etc. can be easily estimated by simple conversion without doing separate numerical simulation for different parameters every time. The analysis of additional parameters like non-dimensional natural frequency can be the input for non-dimensional dynamic analysis. This paper also provides a way for maximization of radial force with the optimized solution of physical dimensions of passive magnetic bearing as inner radius of stator and outer radius of rotor magnet while keeping others as input parameters. The proposed model is validated with data available for radial and axial forces as well as for radial and axial stiffnesses that found in existing literature for similar kind of problem.

Vortex-induced vibrations of a confined circular cylinder for efficient flow power extraction

The effects of confinement on vortex-induced vibration (VIV) of a circular cylinder (diameter D) and its flow power extraction capability are presented. A two-dimensional numerical study is performed on VIV of a circular cylinder inside a parallel plate channel of height H at Reynolds numbers 100 and 150. The blockage ratio (b = D/H) is varied from 1/4 to 1/2. The cylinder is elastically mounted with a spring such that it can only vibrate transversely to the flow. It has a fixed non-dimensional mass (m*) of 10. The energy extraction process is modeled as a damper (damping ratio ζ) attached to the cylinder. With increasing blockage, the vibration amplitude of the cylinder decreases, the lock-in happens at a larger non-dimensional natural frequency of the cylinder, and the initial branch of the vibration response of the cylinder shrinks. There is a minimum value of the channel height and Reynolds number for the existence of the initial branch. The extracted power is found to increase rapidly with the blockage. For the maximum blockage (b = 1/2), the flow power extracted by the cylinder is an order of magnitude larger as compared to what it would extract in an open domain with free stream velocity equal to the channel mean velocity. The optimal mass-damping (α = m*ζ) for extracting the maximum power is found to lie between 0.2 and 0.3. An expression is derived to predict the maximum extracted power from the undamped response of a confined/unconfined cylinder.

Analytical Solution of Sloshing in a Cylindrical Tank with an Elastic Cover

Coupled fluid-structure is significant in many aspects of engineering applications such as aerospace fuel tanks, the seismic safety of storage tanks and tuned liquid dampers. Numerical investigation of the effects of thin plate cover over a cylindrical rigid fuel tank filled by an inviscid, irrotational, and incompressible fluid is investigated. Governing equations of fluid motion coupled by plate vibration are solved analytically. A parameter study on the natural frequency of coupled fluid-structure interaction is performed. The results show the non-dimensional natural frequency of coupled fluid-structure is a function of mass ratio, plate elasticity number and aspect ratio. This function is derived numerically for high aspect ratios which in companion with a semi-analytical could be used in the engineering design of liquid tanks with a cover plate.

Vibration analysis of rotating composite blades with piezoelectric layers in hygrothermal environment

In this study, vibration of a rotating composite blade with piezoelectric layers subjected to a tip mass in hygrothermal environment is investigated. The general composite equations for the single layer materials are expanded under varying temperature and humidity concentrations. The governing equations are derived based on the Hamilton principle based on the Euler-Bernoulli beam theory. Applying Galerkin’s procedure, the resulting equations are converted into a set of eigenvalue equations. The effects of temperature, humidity, angular velocity, fiber orientation angle, voltage and piezoelectric layers on the natural frequency of system are explored. The results show that by increasing the angular velocity, the natural frequencies increases. It was also found that increasing temperature and humidity causes a drop in the non-dimensional natural frequency. Besides, it can be concluded that heat and humidity have significant effects on the natural frequency of rotating composite blades. Applying positive and negative voltages results in a raise and drop in the natural frequency, respectively. It was also found that as the angular velocity of the composite beam increases, the non-dimensional natural frequency decreases. Furthermore, the lowest natural frequency is achieved when the tip mass is located at the end of the beam. The results show that as the fiber orientation angle increases, the natural frequencies increase which is related to enhancing the bending stiffness of the composite blades.

Boundary element analysis of liquid sloshing characteristics in axisymmetric tanks with various porous baffles

This paper aims to investigate the effects of the porous baffles on the suppression of sloshing for the tanks with axisymmetric geometries under lateral excitation. Based on the assumptions of inviscid, irrotational, incompressible liquid and small amplitude sloshing, an axisymmetric boundary element method (BEM) for 3D Laplace equation is derived by using the Green's theorem together with the weighted residual method. And a zoning method is employed to model fluid domain in the tanks with complex porous baffles. Meanwhile, the porous baffles are treated motioning together with the tanks, and the velocity across the porous baffle is assumed to be linearly proportional to the pressure gradient between each side of the porous baffle. And the mechanism of suppressing the sloshing response is mainly the energy dissipation of the fluid passing through the porous baffle. Moreover, the linear free surface boundary conditions are also used to solve the governing equations. Compared with other numerical methods, the most prominent advantage of the BEM in solving axisymmetric potential problem is that only the boundaries of half the cross-section instead of the entire problem domain should be discretized, which can cut down large amount of memory and time costs. The present method is verified by comparing the numerical results with the existing literatures, and excellent agreements are obtained. Meanwhile, the proposed models are applied to investigate the effects of the porous baffles on sloshing response in circular cylindrical, annular cylindrical and conical tanks. The effects of the porous baffle length, porous-effect parameter, installation angle and baffle height on the sloshing force, natural frequency and surface elevation are studied. Additionally, some typical sloshing pressure distributions, velocity potential contours and velocity fields are plotted. The results show that swirls at the tips of the baffles can be observed in many cases, and the top-mounted porous baffle makes more significant suppression effects on sloshing response than that of bottom-mounted porous baffle, while increasing the number of ring porous baffles can achieve better restraint effects on sloshing response. And increasing the baffle length of the horizontal wall-mounted ring porous baffle can significantly decrease the sloshing frequencies, as well as the first non-dimensional natural frequency decreases with decrease in porous-effect parameter of the coaxial porous baffle. In addition, remarkable effects on sloshing can be obtained when reasonable designed by selecting the optimal porous-effect parameter, installation angle and baffle height. And this paper can be a useful guide for the seismic design and analysis of many actual liquid storage tanks (such as the Advanced Passive PWR, large water cooling tower, etc.).

Mode frequency analysis of antisymmetric angle-ply laminated composite stiffened hypar shell with cutout

Abstract This article deals with finite element method for the analysis of antisymmetric angle-ply laminated composite hypar shells (hyperbolic paraboloid bounded by straight edges) that applies an eight-noded isoparametric shell element and a three-noded beam element to study the mode-frequency analysis of stiffened shell with cutout. Two-, 4-, and 10-layered antisymmetric angle-ply laminations with different lamination angles are considered. Among these, 10-layer antisymmetric angle-ply shells are considered for elaborate study. The shells have different boundary conditions along its four edges. The formulation is based on the first-order shear deformation theory. The reduced method of eigen value solution is chosen for the undamped free vibration analysis. The first five modes of natural frequency are presented. The numerical studies are conducted to determine the effects of width-to-thickness ratio (b/h), degree of orthotropy (E 11/E 22), and fiber orientation angle (θ) on the nondimensional natural frequency. The results reveal that free vibration behavior mainly depends on the number of boundary constraints rather than other parametric variations such as change in fiber orientation angle and increase in degree of orthotropy and width-to-thickness ratio.

Analytical analysis for the forced vibration of CNT surrounding elastic medium including thermal effect using nonlocal Euler-Bernoulli theory

This article studies the free and forced vibrations of the carbon nanotubes CNTs embedded in an elastic medium including thermal and dynamic load effects based on nonlocal Euler- Bernoulli beam. A Winkler type elastic foundation is employed to model the interaction of carbon nanotube and the surrounding elastic medium. Influence of all parameters such as nonlocal small-scale effects, high temperature change, Winkler modulus parameter, vibration mode and aspect ratio of short carbon nanotubes on the vibration frequency are analyzed and discussed. The non-local Euler-Bernoulli beam model predicts lower resonance frequencies. The research work reveals the significance of the small-scale coefficient, the vibrational mode number, the elastic medium and the temperature change on the non-dimensional natural frequency.