Quadratic Thickness Research Articles

Non-homogeneity effect on the vibration of the rectangular visco-elastic plate subjected to the linear temperature effect with quadratic thickness variation in both directions

The present work investigates the effect of non-homogeneity on the vibration of a rectangular visco-elastic plate. The plate is subjected to linear temperature variation in the x-direction with quadratic thickness variation in both directions. The quadratic variation has been considered in the material density of the plate only along the x-axis, and it is assumed that non-homogeneity transpires because of this variation. The governing differential equation is solved using the Rayleigh-Ritz technique. All four edges of the plate should be clamped to drive the frequency equation. Deflection and time period have been evaluated for several combinations of values of the thermal constant, constant of nonhomogeneity, taper constants, and length-to-width ratio (aspect ratio) for the first two vibrationmodes of the clamped plate. The results presented are compared with those found in the literature.

Dynamic analysis of heated temperature-dependent bi-directional advanced composites circular plates with quadratic thickness variation

Analysis and numerical results for the axisymmetric vibrations of non-uniform two-dimensional functionally graded circular plates subjected to two-directional temperature distribution are presented on the basis of the exact neutral surface and Mindlin plate theory. The mechanical properties of the plate material are supposed to be temperature-dependent and graded in thickness as well as in radial direction. Based on the thermal boundary conditions, a classical solution of the two-directional heat conduction equation is obtained by the separation of variables method. Equations for thermoelastic equilibrium and axisymmetric motion of the plate are derived in the framework of Hamilton’s principle. The numerical solution of these equations is carried out by the generalized differential quadrature method to compute the thermally induced displacements and natural frequencies using MATLAB. The lowest three values are reported as the frequency parameter for the first three modes of clamped and simply-supported plates. To show the influence of the exact neutral surface, taper parameters, material graded index, heterogeneity and density parameter, numerical results are presented. Mode shapes are plotted for the specified plates.

Splines in vibration analysis of non-homogeneous circular plates of quadratic thickness

Abstract Mathematical model to account for non-homogeneity of plate material is designed, keeping in mind all the physical aspects, and analyzed by applying quintic spline technique for the first time. This method has been applied earlier for other geometry of plates which shows its utility. Accuracy and versatility of the technique are established by comparing with the well-known existing results. Effect of quadratic thickness variation, an exponential variation of non-homogeneity in the radial direction, and variation in density; for the three different outer edge conditions namely clamped, simply supported and free have been computed using MATLAB for the first three modes of vibration. For all the three edge conditions, normalized transverse displacements for a specific plate have been presented which shows the shiftness of nodal radii with the effect of taperness.

Flexural wave control via the profile modulation of non-uniform Timoshenko beams

A methodology to control the flexural wave propagation via modulating the beam profile is presented in this paper. Firstly, the power series method is utilized to theoretically solve the flexural waves in a non-uniform Timoshenko beam with arbitrarily contoured profiles. It is demonstrated that this method is effective for most of inhomogeneous beams as long as the convergence criterion about the beam thickness is satisfied in advance. After validation by comparing the theoretical results with those from the finite element analysis for different thickness variations, lenses are designed by using the quadratic thickness variation pattern with the aim of focusing the wave emitted from a point source on one or two particular positions. Additionally, according to the generalized Snell's law, the thickness-induced phase modulation of the flexural wave is proved from the views of theoretical analysis and numerical simulations, based on which lenses for focusing a plane flexural wave are also realized. It is illustrated through systematic analysis in the frequency domain that all of the lenses designed can exhibit good performances with the evidently increased energy at actual focal positions and the focusing sizes smaller than one working wavelength, which has great potentials in versatile engineering applications.

Flexural waves in a periodic non-uniform Euler-Bernoulli beam: Analysis for arbitrary contour profiles and applications to wave control

The flexural wave in a periodic non-uniform Euler-Bernoulli beam with arbitrarily contoured profiles is studied by utilizing the power series expansion method. The convergence criterion that makes the power series expansion method applicable is also illustrated. The validation is carried out by comparing the theoretical results with that from the finite element analysis when the beam thickness varies in different forms. For a quadratic thickness variation, the first band gap evolution versus the structural parameter is investigated, based on which a flexural-wave-based low-pass filter for frequency shunting and a rectangular lens for energy focusing are designed. It is revealed in the frequency domain analysis that the flexural wave with a lower frequency can propagate further when it travels into the wave filter. The lens designed exhibits a good focusing phenomenon with the focusing size smaller than one wavelength, and has a good performance at a certain finite frequency range. The theoretical method and design scheme can provide effective guidance for the flexural wave control.

Modeling and Simulation of Vibrations of Non-Homogeneous Annular Plate of Quadratic Thickness Resting on Elastic Foundation

Mathematical modeling is presented to analyze natural frequencies of vibrations of an isotropic annular plate of quadratic varying thickness resting on Winkler type elastic foundation where numerical simulation is carried out using quintic spline technique for three different combinations of edge conditions. Effect of elastic foundation, together with nonhomogeneity variation, on the natural frequencies of vibration is illustrated for variety of thickness variation for the first three modes. To compare parametric effect on a specific plate, transverse displacements are presented in normalized form. Accuracy of the results and validity of numerical method is demonstrated by comparing the existing results in the literature.

Thermal effect on radially symmetric vibrations of temperature-dependent FGM circular plates with nonlinear thickness variation

Thermo-elastic analysis for the free axisymmetric vibration of functionally graded circular plates with quadratic thickness variation along radial direction has been presented on the basis of classical theory of plates. Following a power law model, the plate is assumed to be graded in thickness direction and mechanical properties of the material are temperature-dependent. The plate is subjected to a non-linear temperature distribution in thickness direction while the thermal environment over the top and bottom surfaces is uniform. The equations for thermo-elastic equilibrium and axisymmetric motion for such a plate model have been derived using Hamilton’s principle. Employing differential quadrature method, the numerical values of thermal displacements and frequencies for clamped and simply supported plates have been computed. The influence of thickness parameters, material graded index and temperature difference on the vibration characteristics for the first three modes of vibration has been analyzed. The benchmark results for uniform and linear temperature distributions have been computed. A study with the plate material having temperature-independent mechanical properties has also been performed. Because, functionally graded materials inherently withstand high temperature gradients due to low thermal conductivity, core ductility, low thermal expansion coefficient, etc, the performance of results can help for optimal designing of non-uniform circular plates by controlling one or both the taper parameters. In special cases, the results have been compared with those obtained by other methods.

Buckling analysis of circular functionally graded plate under uniform radial compression including shear deformation with linear and quadratic thickness variation on the Pasternak elastic foundation

A numerical scheme for buckling analysis of functionally graded circular plate (FGCP) subjected to uniform radial compression including shear deformation rested on Pasternak elastic foundation is presented. The linear and quadratic thickness variation patterns with various boundary conditions are considered. A modified Euler–Lagrange equation is achieved and then solved by converting differential equation to a nonlinear algebraic system of equations. Also, based on traction–free surface without using shear correction factor, a new approach by considering shear deformation for buckling analysis of FGCP rested on elastic foundation is carried out. The stability equation based on shear stress-free surface is solved by the spectral Ritz method. The spectral Ritz method has good flexibility in the sense of satisfying the boundary conditions. The effects of both linear and quadratic thickness variations and Poisson’s ratio are investigated. By taking small numbers of the basis, the outcomes in literature are improved.

Crashworthiness performance of conical tubes with nonlinear thickness distribution

Crashworthiness performance of conical tubes with various thickness distributions are investigated in the present work. Specimens are fabricated by tube shrinking process and axial crushing tests are carried out to study their energy absorption characteristics. The potential and benefit of tube shrinking to improve crashworthiness performance of tubes are analyzed. Both thickness increase and material hardening during tube shrinking lead to up to 120% increase in energy absorption with even less structural mass. Although the peak force of tubes is increased after shrinking, the load uniformity is still improved due to larger increase in mean crushing force. Numerical simulations are also conducted for tubes with quadratic thickness distributions. The material hardening during fabrication is included and the numerical results compare well with experiment. In addition, crashworthiness optimization of conical tubes with nonlinear thickness distribution is performed by surrogate method. Influences of relevant thickness parameters on performance of the structure are analyzed.

Step index fibre using laser interferometer

A model is suggested to describe the fringe shift which occurs due to the phase variations of cladded glass fibre introduced between the two plates of the liquid wedge interferometer illuminated with a He–Ne laser. The fringe shift of the phase object which appears in the denominator of the Airy distribution formula of the multiple beam interference is represented in the harmonic term. An experiment is conducted using liquid wedge interferometer where the step index glass fibre of a nearly quadratic thickness variation is introduced between the two plates of the interferometer. The obtained fringe shift shows a good agreement with the proposed quadratic model. The Matlab code is written to plot the interferometer fringes comprising the shift of the step index fibre. Secondly, recognition of elliptical fibres is outlined using tomographic imaging. Finally, results and concluding remarks are given.

Thickness-shear modes of an elliptical, contoured at-cut quartz resonator

We study free vibrations of an elliptical crystal resonator of AT-cut quartz with an optimal ratio between the semi-major and semi-minor axes as defined by Mindlin. The resonator is contoured with a quadratic thickness variation. The scalar equation for thickness-shear modes in an AT-cut quartz plate by Tiersten and Smythe is used. Analytical solutions for the frequencies and modes to the scalar equation are obtained using a power series expansion that converges rapidly. The frequencies and modes are exact in the sense that they can satisfy the scalar differential equation and the free edge condition to any desired accuracy. They are simple and can be used conveniently for further studies on other effects on frequencies and modes of contoured resonators.

Thickness-shear vibration of an AT-cut quartz resonator with a hyperbolic contour

A theoretical analysis is performed on thicknessshear vibrations of a contoured AT-cut quartz resonator with a hyperbolic thickness variation using the Legendre equation and hypergeometric function. Based on the solution, resonant frequencies and modes are calculated. Strong energy trapping of the modes is observed. The effects of the parameters of the hyperbolic contour on resonant frequencies and modes are examined. A comparison with the conventional contoured resonator in the literature with a quadratic thickness variation is made. The behaviors of the two types of resonators are qualitatively similar, with small but significant quantitative differences.

Bending analysis of a ceramic-metal arched bridge using a mixed first-order theory

In this research, the bending analysis of an arched bridge is presented based on a mixed first-order thick beam one dimensional plate theory. The present arched bridge is considered as a beam with boundary conditions at its edges, which may be simply-supported, and between these two edges, the beam may have quadratic thickness variation. The bridge consists of two layers; the upper flat one is made from an isotropic homogeneous material such as ceramic, and the lower arched layer is made from an isotropic non-homogeneous functionally graded ceramic-metal material. The upper-surface of the arched layer, which represents the interface between the two layers, is ceramic-rich material while the lower-surface of the arched layer is metal-rich material. This structure eliminates interface problem of the arched bridge and thus the stress distributions are smooth. A closed form solution is developed for the static response of such bridge subjected to different distributed loads. The effects of many parameters on the displacements and stresses are investigated. The sample numerical examples presented herein for bending response of the present arched bridge should serve as references for future comparisons.

Localization of Rayleigh waves in microfluidic channels with quadratic profile

Localization of Rayleigh waves due to mechanical loading of the substrate surface by a liquid layer with quadratic thickness variation in the transverse direction was theoretically studied. Under conditions typical of acoustic biochips, strong localization of the wave field was revealed. This makes it possible to calculate the lowest waveguide modes of the microfluidic channel, neglecting its finite width.

Vibration of Nonhomogeneous Orthotropic Elliptic and Circular Plates With Variable Thickness

In this paper, study of nonhomogeneity as well as variable thickness in elliptic and circular orthotropic plates is undertaken. Nonhomogeneity of plate material is assumed to be a quadratic variation of Young’s modulii and density whereas shear modulus, is considered to vary linearly along both the axes. The quadratic thickness variation in orthotropic nonhomogeneous plates is also considered. Effect of variation of these parameters on vibrational characteristics are analyzed for various boundary conditions at the edges. Results are obtained using boundary characteristic orthogonal polynomials generated by using Gram-Schmidt orthogonalization procedure in Rayleigh-Ritz method.

An exact solution for the bending of thin rectangular plates with uniform, linear, and quadratic thickness variations

In this paper, the analysis of the titled problem is based on classical thin-plate theory, and its numerical solution is carried out by using the small parameter method and Lévy-type approach. The thin rectangular plate considered herein is simply supported on two opposite edges. The boundary conditions at the other two edges may be quite general, and between these two edges the plate may have varying thickness. Closed-form solutions have been developed for the static response of isotropic rectangular plates with non-uniform thickness variation and subjected to arbitrary loading. The accuracy of the present model is demonstrated via problems for which the exact solutions and numerical results are available, and results are also compared with those obtained by using the finite-difference method.

Bending response of a fiber-reinforced viscoelastic arched bridge model

The small parameter approach and the method of effective moduli are presented for the bending response of a fiber-reinforced viscoelastic arched bridge model. The present model is considered as a beam with boundary conditions at its edges, which may be clamped, and between these two edges the beam may have quadratic thickness variation. Closed form solutions have been developed for the quasi-static response of fiber-reinforced viscoelastic arched bridges subjected to uniform loading. The effects due to time, thickness parameter, constitutive parameter, and volume fraction parameter on the deflections are investigated. The sample numerical examples presented herein for bending response of reinforced viscoelastic clamped beams with quadratic thickness variation should serve as references for future comparisons.

The BEM for plates of variable thickness on nonlinear biparametric elastic foundation. An analog equation solution

The BEM is developed for the analysis of plates with variable thickness resting on a nonlinear biparametric elastic foundation. The presented solution is achieved using the Analog Equation Method (AEM). According to the AEM the fourth-order partial differential equation with variable coefficients describing the response of the plate is converted to an equivalent linear problem for a plate with constant stiffness not resting on foundation and subjected only to an `appropriate' fictitious load under the same boundary conditions. The fictitious load is established using a technique based on the BEM and the solution of the actual problem is obtained from the known integral representation of the solution of the substitute problem, which is derived using the static fundamental solution of the biharmonic equation. The method is boundary-only in the sense that the discretization and the integration are performed only on the boundary. To illustrate the method and its efficiency, plates of various shapes are analyzed with linear and quadratic plate thickness variation laws resting on a nonlinear biparametric elastic foundation.

Thermoelastic damping of the low density metals AZ91 and DISPAL

Light metals have high thermoelastic internal friction due to high coefficients of thermal expansion which has to be considered for various structural damping investigations. The thermoelastic effect of a powder metallurgically produced aluminium composite DISPAL™ and a magnesium casting alloy AZ91 (9 wt.% Al, 1 wt.% Zn) was investigated by measurements of the logarithmic decrement of free vibrations of bending beams. Because of inhomogenous stress distribution, a temperature gradient occurs between the compressed and the extended surface of the specimen, which leads to a transverse heat flow. This causes damping that is sensitive to the frequency of vibration. When the time of stress reversal equals the time necessary for heat flow from the compressed to the extended regions, a maximum of the logarithmic decrement occurs. The damping was measured as a function of the specimen thickness and the frequency of vibration ranging from 10 to 130 Hz. The frequency range was realised by both, a variation of the thickness and the mass at the end of the bending beams. By drawing the logarithmic decrement versus the product of the frequency and the quadratic thickness, a Debye peak is obtained which overlays all other damping mechanisms. The thermal diffusivity was calculated by determining the position of the maximum of the peak. The height of the peak, Young's modulus and the specific heat of each sample were used to calculate the coefficient of thermal expansion of the specimens. The values agree, within the limits of scatter, with values measured by a dilatometer and taken from the literature, respectively.

A large-area ionization chamber for portal image calibration

Methods of removing the effects of linear accelerator (linac) output fluctuation from electronic portal images are described and compared. The output of the linac is measured using a specially constructed large-area ionization chamber during imaging and recorded with the image. The use of a dose-rate signal directly from the linac monitor chamber is discussed. Various versions of a quadratic thickness calibration scheme are tested, incorporating linac output data measured by the ionization chamber. Experimental results are presented showing that the incorporation of data from the ionization chamber gives improved absolute calibration accuracy and flatness. Immediately after calibration, the mean systematic thickness error in calibration of a uniform 136.8 mm water-equivalent slab was shown to be no more than 0.6 mm with a thickness variation within each image also of no more than mm. This was true even when imaging with an unstable linac beam giving mean thickness errors between images of 8.8 mm and variations within each image of mm without the ionization chamber correction. Up to one month after calibration, use of the ionization chamber to remove short-term linac fluctuations is shown to still keep mean thickness errors to less than 1.6 mm with variations within each image of no more than mm.