**3. Mathematical modeling**

The mathematical model analyzed is based on the atomic force microscopy system considering a viscoelastic term that is assigned to the medium in which the tip performs the analysis. For this, the proposed model considers the interatomic forces between the probe and the surface of the samples. These interatomic forces are of the van der Waals type that arises from the Lennard-Jones potential. Eqs. (1) and (2) describe the Lennard Jones potential (ULJ) and the van der Waals force (*FWD*), respectively [20, 21].

$$U\_{Lf} = \frac{A\_1 R}{1260z^7} - \frac{A\_2 R}{1260z} \tag{1}$$

$$F\_{WD} = \frac{-\partial U\_{Lf}}{\partial \mathbf{z}} = \frac{A\_1 \mathbf{R}}{180 \mathbf{z}^8} - \frac{A\_2 \mathbf{R}}{6 \mathbf{z}^2} \tag{2}$$

Where *<sup>R</sup>* is the radius of the tip, *<sup>z</sup>* is the distance of the tip, *<sup>A</sup>*<sup>1</sup> <sup>¼</sup> *<sup>π</sup>*<sup>2</sup>*ρ*1*ρ*2*c*1, and *<sup>A</sup>*<sup>2</sup> <sup>¼</sup> *π*<sup>2</sup>*ρ*1*ρ*2*c*<sup>2</sup> are called Hamarke constants (where *ρ*<sup>1</sup> and *ρ*<sup>2</sup> are the number densities of the two interacting kinds of particles, and *c1* and *c2* are the London coefficient). Therefore, the mathematical model analyzed is the one considered by Ref. [20] in which the deflection is determined by the following:

$$w(\mathbf{x}, t) = u(\mathbf{x}, t) + w(\mathbf{x}) + y(t) \tag{3}$$

where *w x*ð Þ , *t* is microcantilever beam deflection, *u x*ð Þ , *t* is a relative deflection of displacement of the actuator, described as *y t*ðÞ¼ *Y* sin ð Þ *Ωt* . We consider the term viscoelastic to the AFM system considering *z* [15, 17]:

$$F\_{\rm CS} = \frac{-\mu\_{\rm eff} b^3 L}{\left[x - w(L) - u(L, \ t) - Y \sin(\Omega t)\right]^3} \dot{u} \tag{4}$$

Where *μeff* is the coefficient of effective viscosity, and *b* and *L* are the width and length of microcantilever, respectively. Considering the vibrations on the intermittent configuration, described by:

$$
\rho A \ddot{u}(\mathbf{x}, t) + EI(u^{\prime\prime\prime}(\mathbf{x}, t) + w^{\prime\prime\prime}(\mathbf{x})) = \left(\frac{-A\_1 R}{180(Z - w(L) - u(L, \ t) - Y \sin\left(\Omega t\right))^3}\right)^2
$$

$$
+ \frac{A\_2 R}{6(Z - w(L) - u(L, \ t) - Y \sin\left(\Omega t\right))^2}
$$

$$
$$

$$
+ \rho A \Omega^2 Y \dot{u}(\mathbf{\Omega t})\tag{5}
$$

Eq. (5) is nonlinear and nonautonomous, and its discretization can be achieved through a dynamic projection on the linear modes of the system. According to Ref. [19], an approximation of the solution of Eq. (5) is by using the linear modes and frequencies of the microcantilever around its electrostatic equilibrium that are different from those of a microcantilever located far from the surface. Therefore,

*MEMS-Based Atomic Force Microscope: Nonlinear Dynamics Analysis and Its Control DOI: http://dx.doi.org/10.5772/intechopen.108880*

calculations of linear modes and microcantilever frequencies over nonlinear electrostatic equilibrium are rigorously calculated using Galerkin's method. The Galerkin's method is used to analyze problems of beams subjected to moving loads with timevarying velocities.

However, we consider under near-resonant forcing, and in the absence of additional internal resonances, only one mode of the microcantilever is assumed to participate in the response.

$$U(\mathbf{x}, t) = \phi\_1(\mathbf{x}) q\_1(t) \tag{6}$$

where *ϕ*1ð Þ *x* is the first approximate eigenfunction about the chosen equilibrium. Substitution of (6) into (5), multiplication of (5) by *ϕ*1ð Þ *x* , subsequent integration over the domain, and the introduction of a modal damping consistent with the Q factors listed in **Table 1** yields the single-degree-of-freedom model:

$$\ddot{\bar{\eta}} = -d\_1 \dot{\bar{\eta}} - \bar{\eta} + B\_1 + \frac{C\_1}{\left(1 - \bar{\eta} - \bar{\eta}\sin\left(\Omega \tau\right)\right)^8} + \frac{C\_2}{\left(1 - \bar{\eta} - \bar{\eta}\sin\left(\Omega \tau\right)\right)^2} \tag{7}$$

$$-\frac{p\dot{\bar{\eta}}}{\left(1 - \bar{\eta} - \bar{\eta}\sin\left(\Omega \tau\right)\right)^3} + \bar{\eta}\bar{\Omega}^2 E\_1 \sin\left(\Omega \tau\right)$$

where : �*<sup>η</sup>* <sup>¼</sup> *<sup>x</sup>*1ð Þ*<sup>τ</sup> <sup>η</sup>* , *x*<sup>1</sup> ¼ *ϕ*1ð Þ *L q*1ð Þ*τ* , *η* ¼ *z* � *wL*, *x*<sup>1</sup> ¼ *ϕ*1ð Þ *L q*1ð Þ*τ* , *η* ¼ *z* � *w L*ð Þ, *<sup>τ</sup>* <sup>¼</sup> *<sup>ω</sup>*1*t*, *<sup>Ω</sup>*� <sup>¼</sup> *<sup>Ω</sup> ω*1 *<sup>d</sup>*<sup>1</sup> <sup>¼</sup> *<sup>C</sup>*<sup>1</sup> *ω*1*ρA* Ð *L* 0 *ϕ*2 1*dx*, *<sup>B</sup>*<sup>1</sup> <sup>¼</sup> ð Þ <sup>1</sup> � *<sup>α</sup>*� *<sup>Γ</sup>*1, *<sup>C</sup>*<sup>1</sup> <sup>¼</sup> *<sup>A</sup>*1*<sup>R</sup>* <sup>180</sup>*k*ð Þ*<sup>η</sup>* <sup>9</sup> *<sup>Γ</sup>*1,*C*<sup>2</sup> <sup>¼</sup> *<sup>A</sup>*2*<sup>R</sup>* <sup>6</sup>*k*ð Þ*<sup>η</sup>* <sup>3</sup> *<sup>Γ</sup>*1, *ω*2 <sup>1</sup> <sup>¼</sup> *EI*<sup>Ð</sup> *<sup>L</sup>* <sup>0</sup> *<sup>ϕ</sup>*1*ϕ*cr <sup>1</sup> *dx ρA* Ð *L* 0 *ϕ*2 <sup>1</sup>*dx* , *<sup>α</sup>*� <sup>¼</sup> *<sup>z</sup> η* , *<sup>Γ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>k</sup>ϕ*<sup>2</sup> <sup>1</sup> ð Þ *L ω*2 <sup>1</sup>*ρA* Ð *L* 0 *ϕ*2 1*dx*, *<sup>k</sup>* <sup>¼</sup> <sup>3</sup>*EI <sup>L</sup>*<sup>3</sup> , *<sup>ζ</sup>* <sup>¼</sup> *<sup>Y</sup> <sup>η</sup>*, *<sup>E</sup>*<sup>1</sup> <sup>¼</sup> *<sup>ϕ</sup>*1ð Þ *<sup>L</sup>* Ð *L* <sup>0</sup> *<sup>ϕ</sup>*1*dx* Ð *L* 0 *ϕ*2 <sup>1</sup>*dx* , and *<sup>p</sup>* <sup>¼</sup> *<sup>μ</sup>eff <sup>b</sup>*<sup>2</sup> *l*.

Thus, considering, �*<sup>η</sup>* � *<sup>x</sup>*<sup>1</sup> and �*η*\_ � *<sup>x</sup>*<sup>2</sup> we can rewrite Eq. (7) in the following system of differential equations:


#### **Table 1.**

*Parameters used for numerical analysis.*

**Figure 2.**

*Scheme of the AFM system. (a) System in initial start and (b) intermittent.*

$$\frac{d\mathbf{x}\_1}{dt} = \mathbf{x}\_2$$

$$\frac{d\mathbf{x}\_2}{dt} = -d\_1\mathbf{x}\_2 - \mathbf{x}\_1 + B\_1 + \frac{C\_1}{\left(1 - \mathbf{x}\_1 - \zeta \sin\left(\varDelta \mathbf{r}\right)\right)^8} + \frac{C\_2}{\left(1 - \mathbf{x}\_1 - \zeta \sin\left(\varDelta \mathbf{r}\right)\right)^2} \tag{8}$$

$$\frac{-p\mathbf{x}\_2}{\left(1 - \mathbf{x}\_1 - \zeta \sin\left(\varDelta \mathbf{r}\right)\right)^3} + \zeta \bar{\mathcal{Q}}^2 E\_1 \sin\left(\varDelta \mathbf{r}\right)$$

**Figure 2** shows the schemes simplify of deflection behavior of nano cantilever of AFM″ for completeness.
