# 2.4.4 Modeling of nonlinear oscillations

Let us study an oscillating cantilever behavior near a sample surface. Presented below is an interactive **Flash model** that allows to investigate changes in the character of resonance and approach-retraction curves depending on the driving force amplitude and probe-sample distance.

*Modeling of interaction force.*

**Fig. 1. The interaction force between probe and sample.**

The interaction force is modeled as follows **(Fig. 1)**:

(1)

which means that just near the sample surface the force is obtained from the **Herz model** taking into account the adhesion in accordance with **DMT model** (adhesive interaction is determined by parameter
). Far enough from the surface, the **Van der Waals forces** dominate. Then, to join the solution, the interaction force between these regions is approximated by the third-order polynomial whose coefficients are dependent on parameters
.

*Calculation of the cantilever parameters.*

**Fig. 2. Schematic geometry of cantilever.**

It is assumed that the rectangular cantilever **(Fig. 2)** with the following sizes:
– beam length,
– width,
– thickness,
– length of the tip, is used in this model. In accordance with the theory (see (12) in **chapter 2.1.2**) the cantilever stiffness is given by

*Calculation of curves.*

The model permits to operate in two modes: study of resonance characteristics (AFC and PFC) and study of approach-retraction curves.

Upon setting the measuring mode of resonance characteristics at a given point above a sample, sets of equations (4), (5) in **chapter 2.4.3** are solved simultaneously **(Fig. 3)**. The obtained curves shape depends much on the driving amplitude. To observe a nonlinear effects (bistability and hysteresis) the cantilever stiffness should be minimum while system Q-factor and driving amplitude should be maximum. Also it is neccesary to choose probe-sample distance short. Otherwise interaction force will be neglect and not affect on resonance characteristics.

**Fig. 3. Screen shot of the flash application. Example of the amplitude-frequency characteristic calculation.**

Upon setting the measuring mode of approach-retraction curves, calculation of oscillations amplitude, resonance frequency (when maximum amplitude is reached) and phase as a function of the probe-sample distance in a given range is performed according to formulas (6), (8), (12) in **chapter 2.4.3**. When several solutions correspond to a certain tip position, small arrows show **(Fig. 4)** what curve will be measured at given scan direction (approach or retraction). Recomendation of parameters choice to obtain nonlinear effects is the same as in case of resonance characteristic.

**Fig. 4. Screen shot of the flash application. Example of the amplitude-distance curve (approach-retraction curve) calculation.**