Geometrically Non-Linear Frequency Response of Axially Functionally Graded Beams Resting on Elastic Foundation Under Harmonic Excitation

Geometrically Non-Linear Frequency Response of Axially Functionally Graded Beams Resting on Elastic Foundation Under Harmonic Excitation

Hareram Lohar (Department of Mechanical Engineering, Jadavpur University, Kolkata, India), Anirban Mitra (Department of Mechanical Engineering, Jadavpur University, Kolkata, India) and Sarmila Sahoo (Department of Civil Engineering, Heritage Institute of Technology, Kolkata, India)
DOI: 10.4018/IJMMME.2018070103
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Abstract

This article presents geometrically nonlinear forced vibration analysis of an axially functionally graded (AFG) non-uniform beam resting on an elastic foundation. The mathematical formulation is displacement based and derivation of governing equations is accomplished following Hamilton's principle. The foundation has been mathematically incorporated into the analysis as a set of linear springs. According to the basic assumption of the present method force equilibrium condition is satisfied at a maximum excitation amplitude value. Thus, the dynamic problem is equivalently represented as a static one, which is solved by following a numerical implementation of the Broyden method. It is a method that utilizes the Jacobian matrix and subsequent correction of the initial Jacobian to solve a system of nonlinear equations. The large amplitude dynamic behaviour of the system in terms of non-dimensional frequency response curves is validated against established results and new results are furnished for a parabolic tapered AFG beam on a linear elastic foundation.
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Introduction

In the recent years it has emerged that pure homogeneous materials, due to certain insufficiency of their properties, are limited in terms of applications in the more technologically advanced domains, such as aerospace industries. New class of advanced materials, namely, composites and functionally graded materials (FGM) have come to the forefront in case of these applications. FGMs are characterized by continuous variation in properties along the spatial directions. Two or more constituent materials are mixed functionally and continuously to obtain FGMs (Koizumi, 1997). These materials provide unique thermo-mechanical properties which are different from the parent materials. It has excellent thermal resistance, low density and high toughness. FGM is also free from interface problems which are severe in case of laminated composite (Nguyen et al., 2013; Suresh and Mortensen, 1998). Due to excellent properties and increasing demand in various areas, a vast body of researchers is devoted to exploring their behavior in different ways.

FG beams can be graded in thickness direction, axial direction and simultaneously along the two directions as well. The thickness directional gradations are most common and have been explored quite comprehensively over the last two decades. Various methodologies like analytical method, semi-analytical method and numerical methods are employed to study the static, dynamic and buckling behavior (Kodali et al., 2008; Yaghoobi and Fereidoon, 2010; Ebrahimi and Salari, 2015; Chen et al., 2015; Hemmatnezhad et al., 2013; Su et al., 2013; Jin and Wang, 2015; Jia et al, 2015; Pradhan and Chakraverty, 2014; Ebrahimi and Zia, 2015) of beams which are graded in the thickness direction. On the other hand, recent years has seen quite a few research works on axially graded beams.

Shahba and Rajasekaran (Shahba and Rajasekaran, 2012) investigated non-uniform Euler–Bernoulli beams with axial inhomogeneity by solving the equations of motion, governing free vibration and stability behaviour of the system. Shahba et al. (Shahba et al., 2011) extended the free vibration and stability analysis to axially graded Timoshenko beams with tapered profile. Li et al. (Li et al., 2013) derived closed form characteristic equations for axially functionally graded (exponential gradation) beams with various end conditions, such as clamped, pinned, guided, free etc., in order to analyse the influence of gradient on frequency spectrum. Kumar et al. (Kumar et al., 2015) employed appropriate energy principles to study free vibration behaviour under loaded and no-load condition of axially graded slender beams with different taper profiles. In this paper authors considered two separate boundary conditions, clamped-clamped and clamped-free. Sarkar and Ganguli (Sarkar and Ganguli, 2014) developed closed form polynomial solutions for a certain class of axially functionally graded uniform Timoshenko beams with fixed–fixed boundary condition, whose governing equations are expressed as coupled differential equations with varying coefficients. Euler Bernoulli theory alongside modified couple stress theory was utilised by Akgöz and Civalek (Akgöz and Civalek, 2013) to present free vibration behaviour of non-homogenous (along the axis) and non-uniform fixed-free microbeams. Huang and Luo (Huang and Luo, 2011) put forward a simple new technique for calculation of critical buckling load of axially arbitrarily inhomogeneous beams with various end conditions and continuous elastic restraint. Huang et al. (Huang et al., 2013) proposed a new approach for exact calculation of natural frequencies of vibration of an AFG non-uniform Timoshenko beam having both ends clamped. A finite element analysis was performed Kien (Kien, 2013) to capture the large amplitude free vibration characteristics of axially functionally graded non-uniform cross-section Timoshenko beams. Zeighampour and Beni (Zeighampour and Beni, 2015) used strain gradient theory to investigate the vibration of axially functionally graded material (AFGM) nanobeam.

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