Nonlinear Oscillations: Exact Solutions and their Approximations

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A review of continuum mechanics models for size-dependent analysis of beams and plates

Nonlinear bending of functionally graded porous micro/nano-beams reinforced with graphene platelets based upon nonlocal strain gradient theory

A first gradient nonlocal model of bending for Timoshenko functionally graded nanobeams based on the Eringen model is proposed. The material properties vary in the thickness direction. A thermodynamic approach is used to provide the variational formulation of the nonlocal model. The governing equations for nanobeam bending with the relevant higher order boundary conditions are then consistently derived. Analytical solutions of the proposed nonlocal functionally graded nanobeam model are provided for a simply supported beam in terms of rotations and transverse displacements. Current results are then compared with existing ones to establish the validity of the present formulation.

Functionally graded Timoshenko nanobeams: A novel nonlocal gradient formulation

Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects

This paper presents a nonlinear vibration analysis of the tensioned nanobeams with simple–simple and clamped–clamped boundary conditions. The size dependent Euler–Bernoulli beam model is applied to tensioned nanobeam. Governing differential equation of motion of the system is obtain by using modified couple stress theory and Hamilton's principle. The small size effect can be obtained by a material length scale parameter. The nonlinear equations of motion including stretching of the neutral axis are derived. Damping and forcing effects are considered in the analysis. The closed form approximate solution of nonlinear equations is solved by using the multiple scale method, a perturbation technique. The frequency-response curves of the system are constructed. Moreover, the effect of different system parameters on the vibration of the system are determined and presented numerically and graphically. The size effect is significant for very thin beams whose height is at the nanoscale. The vibration frequency predicted by the modified couple stress theory is larger than that by the classical beam theory. Comparison studies are also performed to verify the present formulation and solutions.

Size dependent nonlinear vibration of the tensioned nanobeam based on the modified couple stress theory

In this study, nonlinear vibrations of Euler-Bernoulli nanobeams with various supports condition is investigated. The non-linear equations of motion including stretching of the neutral axis are derived. Forcing and damping effects are included in the analysis. Exact solutions for the mode shapes and frequencies are obtained for the linear part of the problem. For the non-linear problem approximate solutions using perturbation technique is applied to the equations of motion. The different of support cases are investigated and the cases analyzed in detail. The method of multiple time scale that is a perturbation technique is applied to the equations of motion. Natural frequencies and mode shapes for the linear problem are found for the nanobeam. Nonlinear frequencies are calculated; amplitude and phase modulation figures are presented for different cases. Frequency-response curves are drawn.

Non-linear vibration of nanobeams with various boundary condition based on nonlocal elasticity theory

In this study, the non-local Euler-Bernoulli beam theory was employed in the nonlinear free and forced vibration analysis of a nanobeam resting on an elastic foundation of the Pasternak type. The analysis considered the effects of the small-scale of the nanobeam on the frequency. By utilizing Hamilton’s principle, the nonlinear equations of motion, including stretching of the neutral axis, are derived. Forcing and damping effects are considered in the analysis. The linear part of the problem is solved by using the first equation of the perturbation series to obtain the natural frequencies. The multiple scale method, a perturbation technique, is applied in order to obtain the approximate closed solution of the nonlinear governing equation. The effects of the various non-local parameters, Winkler and Pasternak parameters, as well as effects of the simple-simple and clamped-clamped boundary conditions on the vibrations, are determined and presented numerically and graphically. The non-local parameter alters the frequency of the nanobeam. Frequency-response curves are drawn.

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Nonlinear Vibration of a Nanobeam on a Pasternak Elastic Foundation Based on Non-Local Euler-Bernoulli Beam Theory

In this study, multi-supported axially moving string is discussed Supports located at the ends of the string are simple supports A support located in the middle section owns the features of a spring String speed is assumed to vary harmonically around an average rate Hamilton’s principle has been used to figure out the nonlinear equations of motion and boundary conditions These equations and boundary conditions are dimensionless Considering the nonlinear effects caused by the string extensions, nonlinear equations of motion are obtained By using multi-timescaled method, which is one of the perturbation methods, approximate solutions have been found The first term in the perturbation series causes the linear problem With the solution of the linear problem, exact natural frequencies have been calculated for different locations of the supports on the middle, various spring coefficients and, with the spring support in the middle of the different location, different spring coefficient and axial speed values Nonlinear terms on second order add correction terms to the linear problem Effect of nonlinear terms on the natural frequency has been calculated for various parameters The cases when the changing frequency of speed is equal to zero, close to zero and close to two times of the natural frequency have been analyzed separately For each case, the stable and unstable areas in the solutions have been identified by stability analysis

Nonlinear vibrations of spring-supported axially moving string

In this study, linear vibrations of an axially moving beam under non-ideal support conditions have been investigated. The main difference of this study from the other studies; the non-ideal clamped support allow minimal rotations and non-ideal simple support carry moment in minimal orders. Axially moving Euler-Bernoulli beam has simple and clamped support conditions that are discussed as combination of ideal and non-ideal boundary with weighting factor (k). Equations of the motion and boundary conditions have been obtained using Hamilton\'s Principle. Method of Multiple Scales, a perturbation technique, has been employed for solving the linear equations of motion. Linear equations of motion are solved and effects of different parameters on natural frequencies are investigated.

Süleyman Murat Bağdatlı

Papers

Size dependent nonlinear vibration of the tensioned nanobeam based on the modified couple stress theory

Non-linear vibration of nanobeams with various boundary condition based on nonlocal elasticity theory

Nonlinear Vibration of a Nanobeam on a Pasternak Elastic Foundation Based on Non-Local Euler-Bernoulli Beam Theory

Nonlinear vibrations of spring-supported axially moving string

Free vibration analysis of axially moving beam under non-ideal conditions