Nonlinear oscillation is an increasingly important and extremely interesting topic in engineering. This article completely reviews a simple method proposed by Ji-Huan He and successfully establishes a fractal undamped Duffing equation through the two-scale fractal derivative in a fractal space. Its variational principle is established, and the two-scale transform method and the fractal frequency formula are adopted to find the approximate frequency of the fractal oscillator. The numerical result shows that He’s frequency formula is a unique tool for the fractal equations.

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The pull-in behavior is an inherent property of the micro-electromechanical oscillator. The bisection method and an iterative method are introduced to find its pull-in voltage, and the main factors affecting the pull-in voltage are elucidated. The simple and efficient operability is demonstrated through theoretical analysis and result comparison.

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In this paper, some approximate analytical methods are reviewed, including the variational iteration method and the homotopy perturbation method. Based on the ideas of the two methods, a new method for solving a class of system of Fredholm integral equations of the second kind is proposed. Some examples are given to show the simple solution process and the accuracy of the solution for each method. The emphasis of this review paper focuses on Ji-Huan He?s unapproachable ideas and the mathematics beauty, so the references are not exhaustive.

The pull-down instability is a newly found phenomenon for oscillators with even nonlinearities. It is a seemingly periodic motion, which will suddenly pull down after some cycles. The phenomenon is similar to the pull-in instability in the micro-electromechanical systems. This paper restudies the well-known Toda oscillator, which is considered to be in periodic motion. Its pseudo-periodic property is studied and the pull-down instability is found for the first time ever. To this end, this paper establishes a variational principle for the Toda oscillator, and the criterion for prediction of the pull-down motion is obtained. A simple formulation to calculate the pull-down instability is proved and verified through examples. Furthermore, an approximate frequency formula is obtained by Li–He modified homotopy perturbation method.

This paper proposes a simple frequency formula developed from He’s frequency formulation for fractal systems. In this approach, the initial guess can be judiciously chosen. Even the simplest initial guess leads to a highly accurate approximate solution. A detailed theoretical development is elucidated, and the solving process is given step by step. The simple calculation and reliable results have been merged into an effective tool for deeply studying fractal vibration systems, and the present approach offers a completely new angle for the fast insight into the physical properties of a non-linear vibration system in a fractal space.