On the geometrically nonlinear vibration of a piezo‐flexomagnetic nanotube

Thermal-mechanical modeling of carbon nanotubes comes with nonlinear equations which are difficult to solve exactly and analytically with any existing general analytical method. However, in many cases of developing symbolic solutions for the nonlinear equations, recourse is made to approximate analytical methods in which their accuracies largely depend on the number of terms included in the solutions. In this work, analytical solutions are presented using differential transformation method with after-treatment technique for nonlinear analysis of thermal and flow-induced vibration of fluid-conveying carbon nanotube resting on Winkler and Pasternak foundations. The developed analytical solutions are used to investigate the effects of nonlocal parameter, Knudsen number, temperature, foundation parameters and boundary conditions on the dynamic behaviour of the single-walled nanotube. Also, the developed analytical solutions are verified with the numerical solutions and validated with experimental results. Good agreements are established between the results of the analytical solutions and the numerical solutions, and also between the results of the analytical solutions and the experimental results. The analytical solutions as presented in this work can serve as benchmarks for other methods of solutions of the problem. They can also provide a starting point for a better understanding of the relationship between the physical quantities of the problems.

Nonlinear thermal and flow-induced vibration analysis of fluid-conveying carbon nanotube resting on Winkler and Pasternak foundations

In the present study, large amplitude free vibration of beams resting on variable elastic foundation is investigated. The Euler–Bernoulli hypothesis and the Winkler model have been applied for beam and elastic foundation, respectively. The beam is axially loaded and is restrained by immovable boundary conditions, which yields stretching during vibrations. The energy method and Hamilton’s principle are used to derive equation of motion, where after decomposition an ordinary differential equation with cubic nonlinear term is obtained. The second order homotopy perturbation method is applied to solve nonlinear equation of motion. An explicit amplitude-frequency relation is achieved from solution with relative error less than 0.07% for all amplitudes. This solution is applied to study effects of variable elastic foundation, amplitude of vibration and axial load on nonlinear frequency of beams with simply supported and fully clamped boundary conditions. Proposed formulation is capable to dealing with any arbitrary distribution of elastic foundation.

Large amplitude free vibration of axially loaded beams resting on variable elastic foundation

IN THIS PAPER, WE PRESENT A NEW NUMERICAL METHOD FOR NONLINEAR VIBRATIONAL ANALYSIS OF EULER-BERNOULLI BEAMS. OUR APPROACH IS BASED ON THE CONTINUOUS GALERKIN-PETROV TIME DISCRETIZATION METHOD. THE EULER-BERNOULLI BEAM EQUATION WHICH GOVERNS ITS VIBRATIONS IS TRANSFORMED INTO SET OF ORDINARY DIFFERENTIAL EQUATIONS AND THE PRESENTED METHOD IS EMPLOYED IN ORDER TO INVESTIGATE THE VIBRATIONAL RESPONSE. A COMPARISON IS MADE BETWEEN PRESENT METHOD AND DIFFERENT OTHER METHODS AVAILABLE IN LITERATURE. IT IS OBSERVED THAT THE OBTAINED RESULTS ARE IN STRONG AGREEMENT WITH OTHER RESULTS IN LITERATURE. WE CONCLUDE THAT THE PRESENT METHOD HAS A GREAT POTENTIAL TO DEAL WITH NONLINEAR VIBRATION ANALYSIS PROBLEMS OF BEAMS AND RELATED STRUCTURES LIKE RODS AND SHAFTS.

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Nonlinear vibration analysis of euler-bernoulli beams by using continuous galerkin-petrov time-discretization method

The paper demonstrates the applicability of a simple intuitive formulation, the development of which started around a decade ago in obtaining the thermal post-buckling behaviour of uniformly heated slender columns that rest on hardening nonlinear Winkler foundation. This foundation, which is firmly bonded to the column, is idealised by a number of closely spaced hardening nonlinear elastic springs. The nonlinear foundation stiffness of the nonlinear elastic foundation is dependent on the square of the deflection of the columns. Analytical expressions for the thermal post-buckling loads, for the buckled mode shapes with wave number one, by applying the concept of smart structures, of the commonly used columns on hardening nonlinear foundation are derived, by using a simple, intuitive formulation. The numerical results presented are validated, by comparing from those obtained for the limiting cases and non-violation of the physical and logical behaviour, in the absence of any similar results available in li...

Applicability of simple intuitive formulation for predicting thermal post-buckling behaviour of columns resting on hardening nonlinear elastic foundation

In this study, a new analytical method called Laplace iteration method (LIM) is used to investigate nonlinear vibration behavior of Euler–Bernoulli and Rayleigh beam on nonlinear elastic foundation with three different combinations of edge conditions (simply supported, clamped–clamped and simply supported–clamped) subjected to axial load. Closed form solutions for natural frequencies, beam deflection, critical buckling and post-buckling loads are also presented. Furthermore, the effects of vibration amplitude, slenderness ratio, linear and nonlinear elastic coefficients of foundation and axial load on the nonlinear frequency and buckling load are discussed. Finally, results obtained by the proposed method are compared with those available in literature. The results showed that in comparison with other approximate analytical solutions, LIM leads to solutions that are in good agreement with respect to existing references which are valid for a wide range of vibration amplitudes meanwhile one or two iterations are enough for obtaining the results.

S. Mehrdad Tabatabaei

Papers

A Novel Approximate Analytical Method for Nonlinear Vibration Analysis of Euler–Bernoulli and Rayleigh Beams on the Nonlinear Elastic Foundation