Zia Saadatnia

Analytical Solutions for Autonomous Conservative Nonlinear Oscillator

This paper adapts the Energy Balance Method (EBM) and Frequency Amplitude Formulation to solve the free vibrations of a conservative oscillator with inertia and static cubic non-linearities. Case studies on the effects of the time response are presented. The results that obtained from the EBM and FAF are then compared with those from the numerical solution in order to verify the accuracy of the proposed method.

Higher order approximate periodic solutions for nonlinear oscillators with the Hamiltonian approach

Abstract In this work, the Hamiltonian approach is applied to obtain the natural frequency of the Duffing oscillator, the nonlinear oscillator with discontinuity and the quintic nonlinear oscillator. The Hamiltonian approach is then extended to the second and third orders to find more precise results. The accuracy of the results obtained is examined through time histories and error analyses for different values for the initial conditions. Excellent agreement of the approximate frequencies and the exact solution is demonstrated. It is shown that this method is powerful and accurate for solving nonlinear conservative oscillatory systems.

Analysis of nonlinear oscillations of a punctual charge in the electric field of a charged ring via a Hamiltonian approach and the energy balance method

Abstract The method of Hamiltonian approach and the energy balance method are applied to obtain the periodic solutions of nonlinear oscillations of a punctual charge in the electric field of charged ring. The obtained approximate frequencies are accurate for the entire range of oscillation amplitudes. A good agreement of the approximate frequencies and periodic solutions with the exact ones are demonstrated and discussed. It is also proved that the results of the energy balance method are better than the Hamiltonian approach for solving this equation.

Application of the Hamiltonian approach to nonlinear oscillators with rational and irrational elastic terms

In this paper, Hamiltonian Approach (HA) is applied to obtain the analytical approximate solution of the nonlinear oscillators with Rational and Irrational Elastic Terms. Periodic solutions are analytically verified and consequently the relationship between the natural frequency and the initial amplitude is obtained in an analytical form. A comparison of the period of oscillation and obtained solutions with the exact results illustrates that the Hamiltonian approach is very effective and quite accurate for nonlinear equations.

Approximate periodic solutions for the Helmholtz-Duffing equation

Approximate periodic solutions for the Helmholtz-Duffing oscillator are obtained in this paper. Hes Energy Balance Method (HEBM) and Hes Frequency Amplitude Formulation (HFAF) are adopted as the solution methods. Oscillation natural frequencies are analytically analyzed. Error analysis is carried out and accuracy of the solution methods is evaluated.

Analytical solution for nonlinear wave propagation in shallow media using the variational iteration method

An analytical solution is presented for nonlinear surface wave propagation. A variational iteration method (VIM) was employed and time-dependent profiles of the surface elevation level and velocity obtained analytically for different initial conditions. It is shown that the VIM used here is a flexible and accurate approach and that it can rapidly converge to the same results obtained by the Adomian decomposition method.

Multi-frequency excitation of microbeams supported by Winkler and Pasternak foundations

The multi-frequency excitation of a microbeam, resting on a nonlinear foundation, is investigated and the governing equation of motion of the microbeam system is developed. The viscoelastic-type foundation is considered by assuming nonlinear parameters for both Pasternak and Winkler coefficients. The well-known Galerkin approach is utilized to discretize the governing equation of motion and to obtain its nonlinear ordinary differential equations. The multiple time-scales method is employed to study the multi-frequency excitation of the microbeam. Furthermore, the resonant conditions due to the external excitation as well as the combination resonances for the first two modes are investigated. The influences of different parameters, namely the Pasternak and Winkler coefficients, the position of the applied force and the geometrical factors on the frequency response of the system are examined.

ANALYTICAL SOLUTIONS FOR NONLINEAR LATERAL SLOSHING IN PARTIALLY- FILLED ELLIPTICAL TANKERS

Dynamic behavior of the large amplitude lateral sloshing is analytically studied in partially filled elliptical tankers. Theory of elliptical trammel pendulums is employed for modeling of the large oscillation of the fluid inside the elliptical container. Nonlinear governing equation of the motion is derived employing the Hamilton principle. Standard and modified Energy Balance Method (EBM) is adopted as the solution technique. Natural frequencies of the free oscillation are analytically obtained as a function of the initial amplitude. It is proved that the nonlinear dynamical system can behave mutually as a hardening and softening system based on the tanker aspect ratio. A number of numerical simulations are carried out and accuracy of the obtained analytical solution is evaluated.

Large Amplitude Free Vibration Analysis of Nanotubes Using Variational and Homotopy Methods

Linear theories are basically unable to model the dynamic behavior of nanotubes due to the large deflection/dimension ratios. In this paper the closed form expressions are obtained for the large-amplitude free vibration of nanotubes. The nonlinear governing differential-integral equation of motion is derived and solved using the Galerkin approach. The derived nonlinear differential equation is then solved using the Variational Approach (VA) and the Homotopy Analysis Method (HAM). The fundamental harmonic as well as higher-order harmonics are analytically obtained. The approximate solutions are compared with those of the numerical responses and accordingly a numerical analysis is carried out. A parametric sensitivity analysis is carried out and different effects of the physical parameters and initial conditions on the natural frequencies are examined. It is found that both the variational analysis and homotopy method are quite consistent and satisfactory techniques to analyze the vibration of nanotubes.Copyright

Nonlinear Forced Vibration Analysis of a Non-Local Carbon Nanotube Carrying Intermediate Mass

The aim of this study is to model and investigate the nonlinear transversal vibration of a carbon nanotube carrying an intermediate mass along the structure considering the nonlocal and non-classical theories. Due to the application of the proposed system in sensors, actuators, mass detection units among others, the analysis of forced vibration of such systems is of an important task being considered here. The governing equation of motion is developed by combining the Euler-Bernoulli beam theory and the Eringen non-local theory. The Galerkin approach is employed to obtain the governing differential equation of the system and the transient beam response for the clamped-hinged boundary condition. A strong perturbation method is utilized to solve the equation obtained and the system responses subjected to a harmonic excitation is examined. The steady-state motion is studied and the frequency response in an analytical form is obtained. Finally, results are evaluated for some numerical parameter values and their effect on the frequency responses are presented and fully discussed.Copyright