Duffing harmonic oscillator is a common model for nonlinear phenomena in science and engineering. Thispaper presents He´ s Energy Balance Method (EBM) for solving nonlinear differential equations. Twostrong nonlinear cases have been studied analytically. The analytical results of the EBM are compared withthe solutions obtained by applying He´ s Frequency Amplitude Formulation (FAF) and numerical solutionsusing Runge-Kutta method. The results show that this method is potentially presented to solve highnonlinear oscillator equations.

Harmonic balance finite element method) HBFEM) is used to estimate the fundamental and harmonic components of magnetic field inside a specimen of ferromagnetic core, when it is subjected to single excitation at its surface. Subsequently iron losses are calculated. For the double excitation, if the signals are incommensurate, then numerical methods fails to yield fundamental and harmonic components of each of signal at various layers. This difficulty has been overcome by the proposed HBFEM with M-functions. The validity of M-functions in respect of nonlinearity of magnetization curve and eddy current loss is verified experimentally.

Axially moving beams are extensively involved in various industries and have significant importance in many mechanical engineering problems. In this paper, the nonlinear forced vibrations of axially moving beam under harmonic force and thermal environment have been studied. In order to considering the effects of transverse shear deformation and rotary inertia, the Timoshenko beam theory has been used to model the axially moving beam. The nonlinear governing equations are derived with the help of Hamilton’s principle. Then the equations and boundary conditions are discretized through generalized differential quadrature method (GDQ) and its differential matrix operators, and accordingly the partial differential equations are converted into the ordinary differential equations. To study the frequency response of the system, the harmonic balance method is used. Also the time responses of the axially moving beam are obtained by the Runge-Kutta method. In a case study, the effects of various parameters such as the axial speed, transverse force acting on the beam, damping coefficient and temperature change on the frequency responses of the axially moving beam with both end simply supported boundary conditions are discussed. The results show that the dynamic behavior of system is significantly affected by any of the mentioned factors.

Nonlinear vibration behavior of beam is an important issue of structural engineering. In this study, a mathematical modeling of a forced nonlinear vibration of Euler-Bernoulli beam resting on nonlinear elastic foundation is presented. The nonlinear vibration behavior of the beam is investigated by using a modified multilevel residue harmonic balance method. The main advantage of the method is that only one nonlinear algebraic equation is generated at each solution level. The computational time of using the new method is much less than that spent on solving the set nonlinear algebraic equations generated in the classical harmonic balance method. Besides the new method can generate higher-level nonlinear solutions neglected by previous multi-level residue harmonic balance methods. The results obtained from the proposed method compared with those obtained by a classical harmonic balance method to verify the accuracy of the method which shows good agreement between the proposed and classical harmonic balance method. Besides, the effect of various parameters such as excitation magnitude, linear and nonlinear foundation stiffness, shearing stiffness etc. on the nonlinear vibration behaviors are examined.

Dynamic behavior of a cracked rotor is simulated under the excitation of unbalance and gravity forces. The shaft is modeled using Timoshenko beam segments supported on two visco elastic bearings, and the attached disks are considered to be solid bodies with point connections to the shaft. The presence of a transverse breathing fatigue crack in the shaft cross section is considered which results in a time dependent stiffness. The model is constructed by using the convenient energy method and the crack effect is implemented as a substructure. Response of the system is calculated using a Harmonic Balance Method (HBM). The calculated system response includes run up response and time response. The obtained results are verified by FEM and a good agreement is observed.

This article analyzes a strongly nonlinear oscillator with cubic and harmonic restoring force and proposes an efficient analytical technique based on the modified energy balance method (MEBM). The proposed method incorporates higher-order approximations. After applying the proposed MEBM, a set of complicated higher-order nonlinear algebraic equations are obtained. Higher-order nonlinear algebraic equations are cumbersome to investigate especially in the case of a large initial oscillation amplitude. This limitation is overcome in the proposed method by using the iterative procedure based on the homotopy perturbation method. The approximated results agree well with the numerically obtained exact solutions. These third-order approximate solutions are found to be almost the same as exact solutions, which cannot be found using the existing energy balance method. Highly accurate result and simple solution procedure are advantages of this proposed method, which could be applied to other nonlinear oscillatory problems arising in nonlinear science and engineering.

In this paper, the harmonic balance technique is used to develop a new computer-aided design method for analyzing nonlinear behavior of log domain filters. Drawing upon the small signal linear model of circuits and power series expansion, previous methods commonly used the second and third terms of the series to analyze distortion. Contrary to conventional linear filters. log-domain filters have large signal behavior and, hence, fail to evaluate their distortions accurately. Compared with the techniques proposed elesewhere, the method proposed in this work, which is based on frequency domain analysis, is not limited by the large signal behavior of log domain filters, and is thus capable of evaluating distortions of log-domain filters more accurately, in a greater number of harmonics, and for circuits with higher complexity levels.

The harmonic projection method can be used to compute some eigenpaires of large matrix, and it is more suitabele for finding interior eigenpairs. However, it has been shown that the harmonic projection method may converge erratically and even may fail to do so. In this paper, we present two new tecniques to improve convergence. The results show that the algorithms converges fast and works with high accuracy.

One of the most variability of atmospheric parameter is precipitation, which changes seasonally. It is important to understand exact characteristic of rainfall seasonal variable. Therefore, it is avoidable to study rainfall season in Hamedan province, because Hameden is one of the important centers of agriculture in Iran. In this research, the seasonal variability of rainfall is considered in Hamedan province. For this object, precipitation data spanning 30 years (1979-2009) were analysed to determine the onset and cessation, trend and seasonal variability of rainfall, based on the daily rainfall for there separated decades. The result of this study show the duration of rainfall is from early falls until end of spring in the first decade. This result indicated the duration of rainfall is long. In the second decade, there is no significant change in rainfall season, but only in some part of the province, the rainfall duration is shortfall and in the other part is long falls. Nevertheless, in the third decade there is considerable movement, i.e. the onset of rainfall duration has been in winter and cessation of rainfall duration has been in summer. It express that rainfall season has moved to summer slowly. The trend of rainfall duration has increasing trend. On the other hand, the duration length of rainfall is longer than before. In order to classify, rainfall seasonal variability is interpolated and the cluster analysis is done base on the “Oghlidos distance” and “Ward” integrated method. According to this method, Hamedan province is classified to 3 groups. The first group indicates the shortfall in the most parts, the second group indicates rainfall season onset is moved to winter, and in the third part the rainfall season hasn’t had change.