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.

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.

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.