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Abstract

In this paper, a comprehensive study is carried out on the dynamic behaviour of Euler–Bernoulli and Timoshenko beams resting on Winkler type variable elastic foundation. The material properties of the beam and the stiffness of the foundation are considered to be varying along the length direction. The free vibration problem is formulated using Rayleigh-Ritz method and Hamilton’s principle is applied to generate the governing equations. The results are presented as non-dimensional natural frequencies for different material gradation models and different foundation stiffness variation models. Two distinct boundary conditions viz., clamped-clamped and simply supported-simply supported are considered in the analysis. The results are validated with existing literature and excellent agreement is observed between the results.

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[29] S.K. Jena, S. Chakraverty, and F. Tornabene. Vibration characteristics of nanobeam with exponentially varying flexural rigidity resting on linearly varying elastic foundation using differential quadrature method. Materials Research Express, 6(8):085051, 2019. doi: 10.1088/2053-1591/ab1f47.
[30] S. Kumar, A. Mitra, and H. Roy. Geometrically nonlinear free vibration analysis of axially functionally graded taper beams. Engineering Science and Technology, an International Journal, 18(4):579–593, 2015. doi: 10.1016/j.jestch.2015.04.003.
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Authors and Affiliations

Saurabh Kumar
1

  1. Department of Mechanical Engineering, School of Engineering, University of Petroleum andEnergy Studies (UPES), Dehradun, 248007, India.
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Abstract

On the basis of Euler-Bernoulli beam theory, the large-amplitude free vibration analysis of functionally graded beams is investigated by means of a finite element formulation. The von Karman type nonlinear strain-displacement relationship is employed where the ends of the beam are constrained to move axially. The material properties are assumed to be graded in the thickness direction according to the power-law and sigmoid distributions. The finite element method is employed to discretize the nonlinear governing equations, which are then solved by the direct numerical integration technique in order to obtain the nonlinear vibration frequencies of functionally graded beams with different boundary conditions. The influences of power-law index, vibration amplitude, beam geometrical parameters and end supports on the free vibration frequencies are studied. The present numerical results compare very well with the results available from the literature where possible.

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Authors and Affiliations

Mehdi Javid
Milad Hemmatnezhad
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Abstract

In this paper, nonlinear free vibration analysis of micro-beams resting on the viscoelastic foundation is investigated by the use of the modified couple stress theory, which is able to capture the size effects for structures in micron and sub-micron scales. To this aim, the gov-erning equation of motion and the boundary conditions are derived using the Euler–Bernoulli beam and the Hamilton’s principle. The Galerkin method is employed to solve the governing nonlinear differential equation and obtain the frequency-amplitude algebraic equation. Final-ly, the effects of different parameters, such as the mode number, aspect ratio of length to height, the normalized length scale parameter and foundation parameters on the natural fre-quency-amplitude curves of doubly simply supported beams are studied.

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Authors and Affiliations

Jafar Eskandari Jam
1
Milad Noorabadi
1
Nader Namdaran
1

  1. Composite Materials and Technology Cente, Malek Ashtar University of Technology, Tehran, Iran

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