A modified generalized, rational harmonic balance method is used to construct approximate frequency-amplitude relations for a conservative nonlinear singular oscillator in which the restoring force is inversely proportional to the dependent variable. The procedure is used to solve the nonlinear differential equation approximately. The approximate frequency obtained using this procedure is more accurate than those obtained using other approximate methods and the discrepancy between the approximate frequency and the exact one is lower than 0.40%.

In this study, the potential use of the Non-Uniform FFT (NUFFT) in SAR imaging is analyzed. The main objective has been the improvement of the computational efficiency and image accuracy of seismic migration SAR processing. Different NUFFT methods have been implemented and tested in order to choose an adequate technique for the imaging problem. Our approach consists in substituting both the Stolt interpolation and the final range inverse FFT, in the ω-k algorithm, by a single NUFFT. Numerical simulations illustrate the performance of the new method and the influence of the selection of NUFFT parameters in the precision and computation time of the SAR imaging algorithm.

An analytical approximate procedure for a class of conservative single degree-of-freedom nonlinear oscillators with odd non-linearity is proposed. This technique is based on the generalized harmonic balance method in which analytical approximate solutions have rational forms. Unlike the classical harmonic balance techniques, in this new procedure the approximate solution and the restoring force are expanded in Fourier series prior to substituting them in the nonlinear differential equation. This approach gives us not only a truly periodic solution but also the frequency of the motion as a function of the amplitude of oscillation. Four nonlinear oscillators are presented to illustrate the usefulness and effectiveness of the proposed technique. The most significant features of this method are its simplicity and its excellent accuracy for the whole range of oscillation amplitude values and the results reveal that this technique is very effective and convenient for solving a class of conservative nonlinear oscillatory systems.

a b s t r a c tA generalized harmonic balance method is used to calculate the periodic solutions of a nonlinear oscillator with discontinuities for which the elastic force term is proportional to sgn(x). This method is a modification of the generalized harmonic balance method in which analytical approximate solutions have rational form. This approach gives us not only a truly periodic solution but also the frequency of the motion as a function of the amplitude of oscillation. We find that this method works very well for the whole range of amplitude of oscillation in the case of the antisymmetric, piecewise constant force oscillator and excellent agreement of the approximate frequencies with the exact one has been demonstrated and discussed. For the second-order approximation we have shown that the relative error in the analytical approximate frequency is 0.24%. We also compared the Fourier series expansions of the analytical approximate solution and the exact one. Comparison of the result obtained using this method with the exact ones reveals that this modified method is very effective and convenient.

A second-order modified rational harmonic balance method is used approximately solve the nonlinear differential equation that governs the oscillations of a conservative autonomous system with one degree of freedom. The Duffing oscillator is analyze to illustrate the usefulness and effectiveness of the proposed technique. We find that this method works very well for this oscillator, and excellent agreement of the approximate frequencies with the exact one has been demonstrated and discussed. For the second-order approximation we have shown that the relative error in the analytical approximate frequency is as low as 0.0055% when A tends to infinity. We also compared the Fourier series expansions of the analytical approximate solution and the exact one. This has allowed us to compare the coefficients for the different harmonic terms in these solutions.

An analytical approximate technique for conservative nonlinear oscillators is proposed. This method is a modification of the generalized harmonic balance method in which analytical approximate solutions have a rational form. This approach gives us not only a truly periodic solution but also the frequency of motion as a function of the amplitude of oscillation. Three truly nonlinear oscillators including the cubic Duffing oscillator, fractional-power restoring force and anti-symmetric quadratic nonlinear oscillators are presented to illustrate the usefulness and effectiveness of the proposed technique. We find that this method works very well for the cubic oscillator, and excellent agreement of the approximate frequencies with the exact one has been demonstrated and discussed. For the second-order approximation, we have shown that the relative error in the analytical approximate frequency is as low as 0.0046%. We also compared the Fourier series expansions of the analytical approximate solution and the exact one. This has allowed us to compare the coefficients for the different harmonic terms in these solutions. For the other two nonlinear oscillators considered, the relative errors in the analytical approximate frequencies are 0.098 and 0.066%, respectively. The most significant features of this method are its simplicity and its excellent accuracy for the whole range of oscillation amplitude values, and the results reveal that this technique is very effective and convenient for solving conservative truly nonlinear oscillatory systems.

This paper presents a new approach for solving accurate approximate analytical solutions for nonlinear phenomena governed by pendulum-like differential equations. The new approach couples Taylor series expansion with rational harmonic balancing. An approximate rational solution depending on a small parameter is considered. After substituting the approximate solution into the governing differential equation, this equation is expanded in Taylor series of the parameter prior to harmonic balancing. The approach gives a cubic equation, which must be solved in order to obtain the value of the small parameter. A method for transforming this cubic equation into a linear equation is presented and discussed. Using this approach, accurate approximate analytical expressions for period and periodic solutions are obtained. We also compared the Fourier series expansions of the analytical approximate solution and the exact one. This allowed us to compare the coefficients for the different harmonic terms in these solutions. These analytical approximations may be of interest for those researchers working in nonlinear physical phenomena governed by pendulum-like differential equations in fields such as classical

An analytical approximate technique for conservative nonlinear oscillators is proposed.This method is a modification of the rational harmonic balance method in which analytical approximate solutions have rational form. This approach gives us the frequency of the motion as a function of the amplitude of oscillation. We find that this method works very well for the whole range of parameters, and excellent agreement of the approximate frequencies with the exact one has been demonstrated and discussed.The most significant features of this method are its simplicity and its excellent accuracy for the whole range of oscillation amplitude values and the results reveal that this technique is very effective and convenient for solving conservative truly nonlinear oscillatory systems with complex nonlinearities.