Free Transverse Research Articles

Mode shapes and frequencies of thin rectangular plates with arbitrarily varying non-homogeneity along two concurrent edges

Analysis and numerical results are presented for free transverse vibrations of isotropic rectangular plates having arbitrarily varying non-homogeneity with the in-plane coordinates along the two concurrent edges on the basis of Kirchhoff plate theory. For the non-homogeneity, a general type of variation for Young’s modulus and density of the plate material has been assumed. Generalized differential quadrature method has been used to obtain the eigenvalue problem for such model of plates for four different combinations of boundary conditions at the edges namely, (i) fully clamped, (ii) two opposite edges are clamped and other two are simply supported, (iii) two opposite edges are clamped and other two are free, and (iv) two opposite edges are simply supported and other two are free. By solving these eigenvalue problems using software MATLAB, the lowest three eigenvalues have been reported as the first three natural frequencies for the first three modes of vibration. The effect of various plate parameters on the vibration characteristics has been analysed. Three dimensional mode shapes have been plotted. A comparison of results with those available in literature has been presented.

Transverse acoustic oscillations in cylindrical chambers with the radial temperature gradient

In this article the transverse acoustic oscillations in cylindrical chambers with a radial temperature gradient are considered. Expression for the velocity potential is calculated. This expression describes the free transverse oscillations in cylindrical chamber with radial temperature gradient.

Free Transverse Vibration Analysis of Axially Functionally Graded Tapered Euler-Bernoulli Beams through Spline Finite Point Method

A new model for the free transverse vibration of axially functionally graded (FG) tapered Euler-Bernoulli beams is developed through the spline finite point method (SFPM) by investigating the effects of the variation of cross-sectional and material properties along the longitudinal directions. In the proposed method, the beam is discretized with a set of uniformly scattered spline nodes along the beam axis instead of meshes, and the displacement field is approximated by the particularly constructed cubic B-spline interpolation functions with good adaptability for various boundary conditions. Unlike traditional discretization and modeling methods, the global structural stiffness and mass matrices for beams of the proposed model are directly generated after spline discretization without needing element meshes, generation, and assembling. The proposed method shows the distinguished features of high modeling efficiency, low computational cost, and convenience for boundary condition treatment. The performance of the proposed method is verified through numerical examples available in the published literature. All results demonstrate that the proposed method can analyze the free vibration of axially FG tapered Euler-Bernoulli beams with various boundary conditions. Moreover, high accuracy and efficiency can be achieved.

MODELING AND VIBRATION ANALYSIS OF COMPLEX ANNULAR SYSTEM WITH USING CYCLIC SYMMETRY FEATURE

In the paper the free transverse vibration of representative toothed gear is considered. Analysed system is treated as an annular plate with complex geometry. The discussion is focused on using the cyclic symmetry modelling technique to vibration analysis of the system under the study. The method of creation of the toothed gear simplified FE model is proposed. The problems related to the determination of critical rotational speeds of the considered system are discussed. Campbell diagram for the studied system is elaborated and the method of designation of the safe working zone of the wheel is presented. Experimental investigations conducted in order to evaluate the proposed method of elaboration of FE models are presented. It is important to note that the issues presented in this work have a practical value for design engineers dealing with the dynamics of rotating systems.

Free vibration analysis of a robotic fish based on a continuous and non-uniform flexible backbone with distributed masses

This paper presents a Differential Quadrature Element Method for free transverse vibration of a robotic fish based on a continuous and non-uniform flexible backbone with distributed masses (fish ribs). The proposed method is based on the theory of a Timoshenko cantilever beam. The effects of the masses (number, magnitude and position) on the value of natural frequencies are investigated. Governing equations, compatibility and boundary conditions are formulated according to the Differential Quadrature rules. The convergence, efficiency and accuracy are compared to other analytical solution proposed in the literature. Moreover, the proposed method has been validate against the physical prototype of a flexible fish backbone. The main advantages of this method, compared to the exact solutions available in the literature are twofold: first, smaller computational cost and second, it allows analysing the free vibration in beams whose section is an arbitrary function, which is normally difficult or even impossible with other analytical methods.

Transverse Vibrations of Mixed-Mode Cracked Nanobeams With Surface Effect

Slant edge cracked effect considering the inherent relation between surface energy and mixed-mode crack propagations on the free transverse vibrations of nanobeams with surface effect is investigated. First, the slant edge cracked effect, which considers residual surface stress effect on the crack tip fields of a mode-I and mode-II surface edge crack, is developed and the corresponding stress intensity factors (SIFs) and local flexibility coefficients are derived. Moreover, a refined continuum model of slant cracked nanobeams is established by considering both slant edge cracked effect and surface effect. The effects of fracture angles, crack depth, surface elasticity, surface stress, and surface density on the local flexibility and free transverse vibration characteristics of cracked nanobeams are, respectively, analyzed. The results show that the flexibility coefficients distribute symmetrically about residual surface stress. Fracture angles have a profound influence on both the symmetries of the mode shapes and the natural frequencies of nanobeams, and the influence becomes more pronounced as crack depth ratios increase. Furthermore, the natural frequencies will first decrease and then increase with fracture angles when the slant edge cracked effect is considered. The results demonstrate that the inherent relation between surface energy and crack propagations should be considered for both the stress distributions at the crack tip and the dynamic behavior of cracked nanobeams.

Free transverse vibration of a wrinkled annular thin film by using finite difference method

This paper investigates the free transverse vibration of a wrinkled annular thin film. The non-dimensional Hamilton motion equation of the wrinkled annular thin film is established, which is solved by using the finite difference method to acquire the vibration frequency and mode. The predicted vibration characteristics are verified by the experimental measurements based on the digital image correlation (DIC) technique. The results show that wrinkles have great effects on the vibration of the annular thin film. Especially for the heavily wrinkled cases, the local–global interactive mode dominates the vibration of the annular thin film. The frequency increases as the wrinkling level increases which is mainly due to the increased nonlinear geometric stiffness. The results provide favorable supports for understanding the role of nonlinear wrinkling on the vibration of thin films.

Simulation of tapered rotating beams with centrifugal stiffening using the Adomian decomposition method

The Adomian modified decomposition method (AMDM) is employed for the free transverse vibration analysis of rotating non-uniform Euler-Bernoulli beams using several boundary conditions, rotation speeds, and beam lengths. The equation of motion includes the axial force due to centrifugal stiffening. The AMDM allows the governing differential equation to become a recursive algebraic equation and after additional simple mathematical operations on the model, the natural frequencies and corresponding closed-form series solution of the mode shape can be obtained simultaneously. The AMDM technique is systematic, so it is straightforward to modify the boundary conditions for specific cases. Comparisons with previously reported results demonstrated that the results obtained by the proposed method were in close agreement.

Weak form quadrature element method for accurate free vibration analysis of thin skew plates

In this paper, a novel weak form quadrature element method (QEM) is proposed for accurate free transverse vibration analysis of thin isotropic skew plates with general boundary conditions. In the formulation of the stiffness matrix, the first and second order derivatives of the shape functions at integration points are computed explicitly by using the differential quadrature rule. This leads to a great simplicity in formulating an N×N-node quadrature plate element and a large reduction of programming effort. Different from the existing weak form quadrature element method or differential quadrature finite element method, the element nodes can be either the same or different from the integration points. Convergence studies are performed. Free vibration of skew thin plates with various skew angles and different combinations of boundary conditions is analyzed. It is shown that although the assumed displacement field does not explicitly consider the bending moment singularities at the obtuse angles, the proposed QEM can yield accurate frequencies even for the thin isotropic skew plate with large skew angles.

Size-dependent bending analysis of FGM nano-sinusoidal plates resting on orthotropic elastic medium

Bending analysis of functionally graded (FG) nano-plates is investigated in the present work based on a new sinusoidal shear deformation theory. The theory accounts for sinusoidal distribution of transverse shear stress, and satisfies the free transverse shear stress conditions on the top and bottom surfaces of the plate without using shear correction factor. The material properties of nano-plate are assumed to vary according to power law distribution of the volume fraction of the constituents. The size effects are considered based on Eringen's nonlocal theory. Governing equations are derived using energy method and Hamilton's principle. The closed-form solutions of simply supported nano-plates are obtained and the results are compared with those of first-order shear deformation theory and higher-order shear deformation theory. The effects of different parameters such as nano-plate length and thickness, elastic foundation, orientation of foundation orthtotropy direction and nonlocal parameters are shown in dimensionless displacement of system. It can be found that with increasing nonlocal parameter, the dimensionless displacement of nano-plate increases.

Free transverse vibration of ocean tower

This paper studies a continuous, elastic model of an ocean tower, partially submerged in water, undergoing free transverse vibration in a plane. The tower is modeled as a non-uniform Timoshenko beam which is supported by an eccentric tip mass on one end and a non-classical damped foundation on the other. The foundation is modeled as a combination of translational and rotational springs and dampers. The effect of shear deformation and rotary inertia is included in the analysis. The free vibration equation is derived using Hamilton׳s variational principle based on two approaches, Rayleigh Ritz Method (RRM) and Finite Element Method (FEM), which show a good agreement in results. The computational efficiency of RRM over FEM is shown using a convergence study. Finally, a parametric study is done to demonstrate the dependence of natural frequency on different configurations of the tower.

On the Transverse, Low Frequency Vibrations of a Traveling String With Boundary Damping

In this paper, we study the free transverse vibrations of an axially moving (gyroscopic) material represented by a perfectly flexible string. The problem can be used as a simple model to describe the low frequency oscillations of elastic structures such as conveyor belts. In order to suppress these oscillations, a spring–mass–dashpot system is attached at the nonfixed end of the string. In this paper, it is assumed that the damping in the dashpot is small and that the axial velocity of the string is small compared to the wave speed of the string. This paper has two main objectives. The first aim is to give explicit approximations of the solution on long timescales by using a multiple-timescales perturbation method. The other goal is to construct accurate approximations of the lower eigenvalues of the problem, which describe the oscillation and the damping properties of the problem. The eigenvalues follow from a so-called characteristic equation obtained by the direct application of the Laplace transform method to the initial-boundary value problem. Both approaches give a complete and accurate picture of the damping and the low frequency oscillatory behavior of the traveling string.

Thermomechanical effects on the bending of antisymmetric cross-ply composite plates using a four variable sinusoidal theory

The thermomechanical bending response of anti-symmetric cross-ply composite plates is investigated by the use of the simple four variable sinusoidal plate theory. The theory accounts for sinusoidal distribution of transverse shear stress, and satisfies the free transverse shear stress conditions on the top and bottom surfaces of the plate without using shear correction factor. By dividing the transverse displacement into bending and shear parts, the number of unknowns and governing equations for the present theory is reduced, significantly facilitating engineering analysis. The validity of the present theory is demonstrated by comparison with solutions available in the literature. Numerical results are presented to demonstrate the behavior of the system. The influences of aspect ratio, side-to-thickness ratio, thermal expansion coefficients ratio and stacking sequence on the thermally induced response are studied. The present study is relevant to aerospace, chemical process and nuclear engineering structures which may be subjected to intense thermal loads.

Hygrothermal vibration of orthotropic double-layered graphene sheets embedded in an elastic medium using the two-variable plate theory

Vibration of orthotropic double-layered graphene sheets under hygrothermal conditions is investigated in this paper using the trigonometric shear deformation plate theory. This theory accounts for sinusoidal distribution of transverse shear stress, and satisfies the free transverse shear stress conditions on the top and bottom surfaces of the plate without using shear correction factor. Unlike the conventional shear deformation theory, the present trigonometric theory contains only two unknowns. The two layers are assumed to be bonded by an internal elastic medium and surrounded by external elastic foundations. The equations of motion are derived based on the nonlocal theory. The motion equations contain the small scale effect as will as hygrothermal effect. The present solution is examined by comparing the present results with those being in the open literature. The effects played by small scale parameter, temperature rise, the degree of moisture concentration, plate aspect ratio and side-to-thickness ratio are studied.

Free vibrations of elastically embedded stocky single-walled carbon nanotubes acted upon by a longitudinally varying magnetic field

The mechanical properties of nano-scaled composites reinforced by carbon nanotubes are enhanced by application of appropriate magnetic fields, however, little is known on the free dynamic response of the magnetically affected stocky single-walled carbon nanotubes (SWCNTs) with elastic supports. Using nonlocal Rayleigh, Timoshenko, and higher-order beam theory, the equations of free transverse vibration of elastically embedded SWCNTs subjected to a longitudinally varying magnetic field are obtained. Since finding an analytical solution to the equations of motion is a very difficult task, an efficient meshless method is proposed. The frequencies of the magnetically affected stocky SWCNTs are evaluated for different boundary conditions. The convergence checks of the proposed numerical models are carried out. In a special case, the obtained results are also compared with those of assumed mode method, and a reasonably good agreement is achieved. Subsequently, the roles of the slenderness ratio of the SWCNT, small-scale parameter, strength of the magnetic field, lateral and rotational interactions of the SWCNT with its surrounding medium on the fundamental frequency are addressed in some detail. The capabilities of the proposed models in capturing the frequencies of the magnetically affected nanostructure are also comprehensively investigated.

Free Vibrations of Beams on Viscoelastic Pasternak Foundations

This paper investigates free transverse vibrations of finite Euler–Bernoulli beams resting on viscoelastic Pasternak foundations. The differential quadrature methods (DQ) are applied directly to the governing equations of the free vibrations. Under the simple supported boundary condition, the natural frequencies of the transverse vibrations are calculated, and compared with the results of the complex mode analysis method. The numerical results obtained by using the DQ and the complex mode methods are in good agreement for the first seven order natural frequencies, but with the growth of the orders, the small quantitative differences between them increase. The effects of the foundation parameters on the natural frequencies are also studied in numerical examples.

Thermal buckling analysis of FG plates resting on elastic foundation based on an efficient and simple trigonometric shear deformation theory

In this paper, an efficient and simple trigonometric shear deformation theory is presented for thermal buckling analysis of functionally graded plates. It is assumed that the plate is in contact with elastic foundation during deformation. The theory accounts for sinusoidal distribution of transverse shear stress, and satisfies the free transverse shear stress conditions on the top and bottom surfaces of the plate without using shear correction factor. Unlike the conventional trigonometric shear deformation theory, the proposed sinusoidal shear deformation theory contains only four unknowns. It is assumed that the mechanical and thermal non-homogeneous properties of functionally graded plate vary smoothly by distribution of power law across the plate thickness. Using the non-linear strain-displacement relations, the equilibrium and stability equations of plates made of functionally graded materials are derived. The boundary conditions for the plate are assumed to be simply supported on all edges. The elastic foundation is modelled by two-parameters Pasternak model, which is obtained by adding a shear layer to the Winkler model. The effects of thermal loading types and variations of power of functionally graded material, aspect ratio, and thickness ratio on the critical buckling temperature of functionally graded plates are investigated and discussed.

Transverse Musculocutaneous Gracilis Flap for Treatment of Capsular Contracture in Tertiary Breast Reconstruction

Capsular contracture is a common complication associated with implant-based breast reconstruction and augmentation leading to pain, displacement, and rupture. After capsulectomy and implant exchange, the problem often reappears. We performed 52 deepithelialized free transverse musculocutaneous gracilis (TMG) flaps in 33 patients for tertiary breast reconstruction or augmentation of small- and medium-sized breasts. The indications for implant removal were unnatural feel and emotion of their breasts with foreign body feel, asymmetry, pain, and sensation of cold. Anyway, most of the patients did not have a severe capsular contracture deformity. The TMG flap is formed into a cone shape by bringing the tips of the ellipse together. Depending on the contralateral breast, the muscle can also be shaped in an S-form to get more projection if needed. The operating time for unilateral TMG flap breast reconstruction or augmentation was on average 3 hours and for bilateral procedure 5 hours. One patient had a secondary revision of the donor site due to disruption of the normal gluteal fold. Eighty percent of the unilateral TMG flap reconstructions had a lipofilling procedure afterward to correct small irregularities or asymmetry. The advantages of the TMG flap such as short harvesting time, inconspicuous donor site, and the possibility of having a natural breast shape make it our first choice to treat capsular contracture after breast reconstruction and augmentation.

Dynamic interactions of doubly orthogonal stocky single-walled carbon nanotubes

Nonlocal continuum-based modeling of free transverse vibration of doubly orthogonal stocky single-walled carbon nanotubes (DOSWCNTs) which are embedded in an elastic matrix is of particular interest. A basic step towards such a crucial goal is to establish a rational model for evaluating the van der Waals interactional force between two tubes. By employing Hamilton’s principle, the nonlocal governing equations associated with the free vibration of the nanosystem are obtained using Timoshenko and higher-order beam theories. Inherently, these are sets of four coupled integro-partial differential equations which are very difficult to be solved analytically. To bridge such a dilemma, reproducing kernel particle method is exploited and the essential boundary conditions of each shear-deformable beam model are exactly satisfied by implementing the corrected collocation method. In a special case, the predicted natural frequencies by the proposed models are checked with those of other models analyzed via mode summation method, and a reasonably good agreement is achieved. Subsequently, the effects of the slenderness ratio, small-scale parameter, aspect ratio, nanotubes’ radii, intertube distance, transverse and rotational stiffness of the surrounding elastic medium on free vibration of the nanosystem are comprehensively addressed for various boundary conditions.

3-node Basic Displacement Functions in Analysis of Non-Prismatic Beams

Purpose– Analysis of non-prismatic beams has been focused of attention due to wide use in complex structures such as aircraft, turbine blades and space vehicles. Apart from aesthetic aspect, optimization of strength and weight is achieved in use of this type of structures. The purpose of this paper is to present new shape functions, namely 3-node Basic Displacement Functions (BDFs) for derivation of structural matrices for general non-prismatic Euler-Bernoulli beam elements. Design/methodology/approach– Static analysis and free transverse vibration of non-prismatic beams are extracted studied from a mechanical point of view. Following structural/mechanical principles, new static shape functions are in terms of BDFs, which are obtained using unit-dummy-load method. All types of cross-sections and cross-sectional dimensions of the beam element could be considered in this method. Findings– According to the outcome of static analysis, it is verified that exact results are obtained by applying one or a few elements. Furthermore, it is observed that results from both static and free transverse vibration analysis are in good agreement with the previous published once in the literature. Research limitations/implications– The method can be extended to structural analysis of curved and Timoshenko beams as well as plates and shells. Furthermore, exact dynamic shape functions can be derived using BDFs by solving the governing equation for transverse vibration of beams. Originality/value– The present investigation introduces new shape functions, namely 3-node Basic Displacement Functions (BDFs) extended from 2-node functions, and then compares its performance with previous element.