I'm now styduing micro-cantilever used as a sensor. It is usually made of silicon based materials. In a certain condition below, the cantilever will vibrate itself when a certain external frequency is applied. <condition> Certain cantilevers may exhibit mode coupling depending on the device characteristics. Analytical expressions for various mode shapes can be derived mathematically from the corresponding equations of motion under the following assumptions: the aspect ratio is sufficiently large, the deflection is small compared to cantilever thickness, the geometry is of single-layer uniform rectangular cross-section, and the material is isotropic. There are four types of vibrating modes. Out-of plane vibrations include transverse, also called bending or flexural, and torsional motion. In-plane vibrations include lateral, also called in-plane bending, and longitudinal, also called extensional or axial, motion. Each of the four modes exhibit resonance when excited at their characteristic frequency, known as the resonant frequency or eigen frequency. My question is why a microcantilever has different resonant modes in different frequencies on a single material. And how can the torsional mode be occurred? It's very interesting.