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In this study, we investigate the buckling of a column made of functionally graded materials in axial direction. The column is simply supported at both ends. The mode shape is represented by a polynomial whose power is increased from four to nine; this power only restricted by the possibility of the hand- or symbolic-manipulation. It is shown that the solutions stemming from lower-order polynomial are all contained in the solution associated with higher-order polynomials. Numerous novel solutions are uncovered that could be used as benchmark solutions versus numerous approximate techniques available for eigenvalue problems. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we present new closed‐form solutions for functionally graded longitudinally vibrating bar. The modulus of elasticity is represented as a continuous function varying along the axial coordinate. It is postulated that the mode shape represents the sixth order polynomial. Six closed form solutions are presented: four for clamped‐clamped bar, and two for clamped‐free bar. [ABSTRACT FROM AUTHOR]

In this paper, we present a general methodology for solving buckling problems for inhomogeneous columns. Columns that are treated are functionally graded in axial direction. The buckling mode is postulated as the general order polynomial function that satisfies all boundary conditions. For specificity, we concentrate on the boundary conditions of simple support, and employ the second-order ordinary differential equation that governs the buckling behavior. A quadratic polynomial is adopted for the description of the column's flexural rigidity. Satisfaction of the governing differential equation leads to a set of nonlinear algebraic equations that are solved exactly. In addition to the recovery of the solutions previously found by Duncan and Elishakoff, several new solutions are arrived at. [ABSTRACT FROM AUTHOR]

In this study we contrast two competing methodologies for the impact buckling of a column that is clamped at both ends. The initial imperfection is postulated to be co-conflgurational with the fundamental mode shape of the column without the axial loading. A solution is also furnished for the case when the initial impeifection is proportional to the Filonenko-Borodich "cosinusoidal polynomial". Probabilistic and interval analyses are conducted for each case; these are contrasted on some representative numerical data. [ABSTRACT FROM AUTHOR]

Buckling of a column made of functionally graded material is investigated. The functional grading is performed in the longitudinal direction. The following problem is addressed: determine the variation of the elastic modulus with the axial coordinate such that the buckling load exceeds the preselected value. In this study, only polynomial variation of the buckling mode is considered and only the case of cantilever column is treated. [ABSTRACT FROM AUTHOR]

Natural frequencies of triple-walled carbon nanotubes (TWCNTs) are determined both exactly and approximately. For the case of a TWCNT simply supported at its ends, closed-form solutions are obtained. It is shown that there are three series of natural frequencies corresponding to the cubic polynomial equation for natural frequency squares. For a TWCNT that has other boundary conditions approximate Bubnov-Galerkin and Petrov-Galerkin methods are used with simple polynomial functions. Each of these methods yields three natural frequencies corresponding to the lower end of each frequency spectrum. [ABSTRACT FROM AUTHOR]

Abstract: The titled problem is solved by the aid of the semi-inverse formulation. At the first glance, a surprising demand is posed: Design a radially heterogeneous polar orthotropic circular plate whose fundamental mode shape coalesces with the static deflection of a homogeneous circular plate of the same radius and thickness under uniformly distributed load. Compatible polynomial variations of the radial and circumferential flexural rigidities are introduced. It turns out that these are infinite amount of polar orthotropic circular plates that solve the posed semi-inverse problem, depending on the free parameter represented by one of the coefficients in variation of flexural rigidities. This allows to provide an unique solution to the titled problem: there is a single plate that possesses the specified fundamental natural frequency. [Copyright &y& Elsevier]

In this study, the vibration tailoring problem is analytically solved for the polar orthotropic circular plate with translational spring along its circumference. By using the semi-inverse method and postulating the mode shape as a polynomial, we derive a closed-form solution. [ABSTRACT FROM AUTHOR]

Abstract: The semi-inverse problem of free vibration of simply supported inhomogeneous polar orthotropic plates is studied. The mode shape is postulated in the form of the classic formula by Lekhnitskii, namely, the static deflection of the associated uniform polar orthotropic circular plate under uniform loading. The ratios of the circumferential rigidity to the radial rigidity are identified as integer numbers, so that the candidate mode shapes constitute the polynomial functions. The flexural rigidities themselves are also represented by polynomials. The semi-inverse problem of identifying the coefficients of the flexural rigidities is solved analytically. It appears remarkable, that for numerous cases the simple method developed in this study provides novel closed form solutions for the design of the polar orthotropic circular plates with pre-specified mode shapes. [Copyright &y& Elsevier]

Abstract: Apparently for the first time in the literature, an exact closed-form solution is derived for an axisymmetrically vibrating inhomogeneous circular plate that is simply supported at its boundary. The solution is characterized here as an unusual one since for its counterpart—the homogeneous plate, transcendental functions are called for whereas here a solution is found in elementary functions, namely, polynomials. The analysis is based on an inverse vibration problem: Given a candidate mode shape and density distribution, we calculate the plate stiffness so that the governing equation for the plate mode shape is identically satisfied. We also are able to obtain the expression for the corresponding natural frequency. [Copyright &y& Elsevier]

Abstract: In this paper, we use the semi-inverse method to find the solutions to the dynamic equation of inhomogeneous, functionally graded simply supported beams. For the anti-symmetric mode and the material density (or the Young''s modulus) that are both polynomial functions and and have been pre-specified, we find the Young''s modulus (or the density) in polynomial functions and determine the closed from expression for the natural frequency. [Copyright &y& Elsevier]