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Many nonlinear systems in industry including oscillators can be simulated as a mass-spring system. In reality, all kinds of oscillators are nonlinear due to the nonlinear nature of springs. Due to this nonlinearity, most of the studies on oscillation systems are numerically carried out while an analytical approach with a closed form expression for system response would be very useful in different applications. Some analytical techniques have been presented in the literature for the solution of strong nonlinear oscillators as well as approximate and numerical solutions. In this paper, Amplitude-Frequency Formulation (AFF) approach is applied to analyze some periodic problems arising in classical dynamics. Results are compared with another approximate analytical technique called Energy Balance Method developed by the authors (EBM) and also numerical solutions. Close agreement of the obtained results reveal the accuracy of the employed method for several practical problems in engineering.

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The behavior of ascending bubbles is of fundamental importance for understanding many natural and artificial phenomena. In this study, the unsteady motion of a single spherical air bubble rising in a stationary viscous liquid has been studied for low Reynolds numbers. Bubble rising under the in?uence of gravitational force is one of the most common gas–liquid flow phenomena. Understanding the dynamic interaction between the phases is an important key for the design and operation of industrial applications. A particular instance of this situation occurs in gas–liquid column reactors. The equation of motion included an additional mass term and neglected the Basset term. An exact solution is derived for instantaneous velocity using a series‐based technique, called the differential transformation method (DTM), and the acceleration and position of the bubble were determined. A general solution of the equation was obtained, and also for several practical conditions with air as the gas phase and glycerin solutions of different concentrations as liquid phase. The results obtained demonstrate the effectiveness of using DTM and promote a new application of this powerful analytical tool for solving nonlinear problems in two‐phase flows. Copyright © 2010 Curtin University of Technology and John Wiley & Sons, Ltd.