In this paper, we investigate the mechanism of atomic force microscopy in tapping mode (AFM-TM) under the Casimir and van der Waals (VdW) forces. The dynamic behavior of the system is analyzed through a nonlinear dimensionless mathematical model. Numerical tools as Poincaré maps, Lyapunov exponents, and bifurcation diagrams are accounted for the analysis of the system. With that, the regions in which the system presents chaotic and periodic behaviors are obtained and investigated. Moreover, the fractional calculus is introduced into the mathematical model, employing the Riemann-Liouville kernel discretization in the viscoelastic term of the system. The 0-1 test is implemented to analyze the new dynamics of the system, allowing the identification of the chaotic and periodic regimes of the AFM system. The dynamic results of the conventional (integer derivative) and fractional models reveal the need for the application of control techniques such as Optimum Linear Feedback Control (OLFC), State-Dependent Riccati Equations (SDRE) by using feedback control, and the Time-Delayed Feedback Control. The results of the control techniques are efficient with and without the fractional-order derivative.

Technological advances in the development of electromechanical systems are gaining ground in the most diverse branches of engineering science. Such advances permit the development of smaller devices that vary from macro- to nanoscale, which have opened space for new research fields. However, the size scale of these devices has been a challenge as classical mechanics is not the only applicable one anymore. Mainly, nanoelectromechanical systems (NEMS) are affected by quantum forces, whose systems have been extensively studied in the past years [

A special mechanism has been used for sample surfaces analysis at atomic scale, which is the atomic force microscope characterized as a NEMS, mostly referred to as atomic force microscopy technique. This technique is very well established as it is a very precise superficial analysis and has allowed the increase of the understanding and analysis of very small and soft materials such as polymeric materials [

Among the AFM in tapping mode (AFM-TM), contact and noncontact with the sample surface stand out, as they can form a three-dimensional image of such surface [

The van der Waals (VdW) forces are the most dominant atomic force in the AFM-TM and can be found in various works in the literature as in [

The VdW forces consider the electron density fluctuations present between the test tip atoms and the surface. In [

On the other hand, different quantic forces can become as dominant as VdW depending on the material of the surface to be scanned by the AFM mechanism. The authors in [

The intersection of the transition between the Casimir and VdW forces is also discussed in [

The atomic forces have been under study and are of great interest due to their effect on the dynamics of the microcantilever beam of the AFM. The mechanism becomes inoperable due to irregular measurements as mentioned in [

In this work, we approach computationally the dynamics and control of the AFM-TM model proposed by [

There is an analysis of the nanosystem for a particular case where there is the coexistence of both Casimir and VdW forces in the system. Since the Casimir force is a problem to be solved yet, it is considered as a function and only numerical simulations of the system are carried out. Consolidated numerical techniques as Poincaré maps, Lyapunov exponent calculus, and bifurcation diagram are carried out for the system [

In addition, squeeze-film damping is also introduced due to the small distance between the microcantilever and surface as a viscoelastic term. The viscoelastic term is an approximation for the behavior of the indented analysis tip during the tapping process, commonly observed in the analysis of biological samples that, due to high vibrations and small distances, a gas film is generated [

In both fractional model and nonfractional models, the chaotic behavior of the system is analyzed and control techniques are proposed in order to control the chaotic behavior in the dynamics of the AFM system, which are Optimum Linear Feedback Control (OLFC) and State-Dependent Riccati Equations (SDRE) feedback controls and the Time-Delayed Feedback Control (TDFC).

The rest of the paper is organized as follows. Section

Figures _{0} is the distance between the equilibrium point of the microcantilever and the analyzed sample, _{0}) is the Lennard-Jones potential, _{1} is the Hamaker constant to the attractive potential, and _{2} is the Hamaker constant to the repulsive potential.

Representation of AFM-TM schematic diagram. (a) Representation of microcantilever beam and (b) representation of analysis tip through a mass-spring-damper model.

The van der Waals forces is described as a combination of the attractive and the repulsive parcels yielding

As the AFM-TM analysis is carried out on tapping mode, there is an excitation force on the microcantilever by the piezoelectric actuator given by _{0}cos (_{VdW} is the attraction and repulsive forces described by (_{k}, the structural damping force is _{c}, the squeeze-film damping is given by _{cs}, and the Casimir force is _{cas}.

The conservative force of the spring _{k} is given by_{c}_{d} is the structural damping coefficient of the microcantilever. The damping force generated by the squeeze-film damping _{cs} is denoted by_{eff} is the coefficient of effective viscosity and

The Casimir force _{cas} is given by

Based on the forces described in (

Carrying out a dimensionless procedure into (_{1} is the frequency of the first mode of vibration of the microcantilever and

Table

Parameters of the AFM-TM model [

Description | Value |
---|---|

Length of the cantilever | 449 |

Width of the cantilever | 46 |

Thickness of the cantilever | 1 |

Tip radius | 0.15 |

Material density | 2,330 kg/m^{3} |

Young’s modulus | 176 GPa |

Bending stiffness | 0^{−1} |

Frequency of the 1st mode of the cantilever | 11804 kHz |

Quality factor | 100 |

Hamaker constant (repulsive) | 1^{−70} J⋅m^{6} |

Hamaker constant (attractive) | 1^{−19} J |

Using Table ^{st} mode of vibration (Ω = 1.0) [

In this subsection, the dynamic behavior influenced by the squeeze-film damping and the Casimir force is investigated. Numerical simulations are carried out by using the 4th order Runge-Kutta implicit method with integration step ^{r}) [

The squeeze-film damping is inherent to the sample, providing a nonlinear damping. However, the Casimir force is only presented at nanodistances [

Figure

Lyapunov exponent in 2D with

It is observed that, in the region of the parameters (

Figures

(a) Bifurcation diagram and (b) Lyapunov exponent with

Figures

(a) Poincaré map and (b) phase planet with

For a temporal view of the chaotic behavior for

(a) Time history of displacement _{1} and (b) time history of velocity _{2} with

With the previous analyses of the system that showed the presence of chaotic behavior for parameters

As originally suggested by the author in [

Assuming that the velocity of oscillation is measured as an output of the nonlinear system (

The NEMS system with the control signal of (

The time delay

Bifurcation diagram for control gain

For any

Time histories of (a) displacement and (b) velocity; (c) phase portrait of the system with controlled trajectory (solid line) and the trajectory without control (dashed line); (d) control signal

The obtained periodic orbits with the time delay control of Figure

The time-delayed control was efficient in leading the system to a periodic orbit. As can be seen in the results, the control only used the control signal when necessary to take the system to a periodic orbit of the system, as observed in Figure

However, for the cases where it is desired to take the system to a previously defined orbit, the control may not be the most suitable. For these mentioned cases, the optimal control is considered in the next section.

In the following, the solution of these problems using the optimal linear control technique for nonlinear systems developed by the authors in [

However, the vector control

Substituting (

The system can be represented in the following form:

The optimal linear feedback control is applied as it was introduced by the authors in [

According to [

Minimizing the cost functional to

The control

Having

The matrices

Solving (

Substituting the matrices

In Figure

(a) Displacement of the tip with control (solid line) and without control (dashed line); (b) phase portrait of the tip with control (solid line) and without control (dashed line); (c) error variation

According to Figure

For the SDRE control, (

Considering that, the matrices

In addition, by definition

The quadratic performance measured for the feedback control problem is given by

Having

Another important factor to consider is that the matrix

Then, to obtain a suboptimal solution for the dynamic control problem, the SDRE technique has the following procedure:

Define the state-space model with the state-dependent coefficient as in (

Define _{0}, so that the rank of

Solve Riccati equation (

Calculate the input signal from (

Integrate system (

Calculate the rank of (

Figure

(a) Displacement of the tip with control (solid line) and without control (dashed line); (b) phase portrait of the tip with control (solid line) and without control (dashed line); (c) error variation

The optimal linear feedback control by the SDRE is demonstrated to be effective in leading and maintaining the system in the desired orbit. Comparing the results obtained with the OLFC and SDRE controls, the results similarity is evident. The reason for that is because both controls are obtained considering an optimal control strategy.

To analyze these controls in more detail, the next section presents a study of the parametric sensitivity of the controls.

The control design is usually based on parameters of the mathematical model obtained from physical laws governed by the dynamic behavior of the system. Often, because of the limitations of the knowledge process, the used models do not accurately represent the real dynamics. Consequently, the control design cannot operate as intended when applied in a real process, because the parameters used in the control may contain parametric errors. To solve this problem, many researches have focused on incorporating the uncertainties associated with real structures into numerical simulation for reliable predictions [

To consider the effects of parameter uncertainties on the performance of the controller, the parameters used for the controls are considered as a random error of

Figure

Deviation of the desired trajectory with the proposed control with parametric errors only in feedback control _{1}; (b) deviation of the desired trajectory _{2}.

Figures

Deviation of the desired trajectory with the proposed control with parametric errors only in feedback control _{1}; (b) deviation of the desired trajectory _{2}.

In Figures

Deviation of the desired trajectory with the proposed control with parametric errors in feedback control (_{1}; (b) deviation of the desired trajectory _{2}.

In Figures

Deviation of the desired trajectory with the proposed control with parametric errors in feedback control _{1}; (b) deviation of the desired trajectory _{2}.

Phase planes of the system in fractional-order derivative equations: (a) _{3} = 1.0 and (b) _{3} = 0.5031.

As can be seen in Figures

Due to the small scale of the AFM operation, a viscoelastic behavior of the system is observed. For the viscoelastic behavior, it is possible to use the fractional-order derivatives to analyze the dynamic behavior of the system [_{0} = 0, 0 <

Hence, the technique of fractional calculus is included to analyze the behavior of system (

Figures _{3} = 1.0 and _{3} = 0.5031, respectively. In addition, Figure _{3} = 1.0 and _{3} = 0.5031. Those results show an evidence of irregular motion, which is an evidence of chaotic behavior.

Time series of fractional order of the system with _{3} = 1.0 and _{3} = 0.5031.

To calculate the Lyapunov exponent for system (

The 0-1 test, proposed by the authors in [_{1} of (_{c}. The test considers a system variable _{j}, where two new coordinates (_{c} in the limit of a long time

Given any two vectors _{c} value is close to 1, the system is chaotic.

Figure _{c} of the 0-1 test for a scan of the fractional derivative _{3} mainly to _{c} is very close to 1.

Representation of Kc (0-1 test) vs.

As for

In this section, the techniques of fractional calculus to analyze the behavior of system (

The vector control

In this way, the desired regime is obtained by

The feedforward control

Substituting (

The system can be represented in the following form:

Since the objective of this work is to control _{1} and _{2}, the variable _{3} is considered only as a disturbance of the system, as similarly proposed in [

The matrices

Figure

OLFC control: (a) phase portrait with control (solid line) and without control (dashed line); (b) trajectory of the control (dashed line) in comparison with the desired trajectory (solid line); (c) error variation

The SDRE control considers the matrices

Figure

SDRE control: (a) phase portrait with control (solid line) and without control (dashed line); (b) trajectory of the control (dashed line) in comparison with the desired trajectory (solid line); (c) error variation

As observed in Figures

Both OLFC and SDRE control techniques are very efficient in controlling the chaotic AFM system which presented chaotic behavior due to the presence of squeeze-film damping considered as a viscous damping ((

This work presented the dynamical analysis and control of an AFM-TM system with the addition of the Casimir force and VdW forces that induced the presence of a chaotic behavior. In addition, the investigation of the fractional-order derivative is sought out to allude the influence of the viscoelastic term in the AFM-TM setup. A set of parameters enunciated the chaotic behavior. Thus, the intensity of the Casimir force strongly influenced the behavior of the AFM-TM, differently of the damping parameter

In order to suppress the chaotic motion, the TDFC and the optimal control by OLFC and SDRE techniques were projected. The efficiency of the proposed techniques was demonstrated through numerical simulations. As could be seen in Figure

As the time-delayed control is not designed to take the system to any previously defined orbit, an alternative is the application of the OLFC or SDRE control. As could be seen in Figures

Thus, it is possible to conclude that the TDFC is an excellent option when the objective is to take the system to a periodic orbit with the lowest cost of control. The OLFC or SDRE control is a viable option to be considered in cases in which it is necessary to impose the desired orbit. In addition, the robustness of the OLFC and SDRE controls was analyzed. Both OLFC and SDRE techniques worked well and were robust due to parametric error analyses.

As the OLFC and SDRE control proved to be robust to parametric errors, the application of the control in the fractional-order system was also considered. Numerical results showed that the OLFC and SDRE controls are also effective for control in fractional-order systems. For the OLFC control, the feedback control is linear and not state dependent, so its processing is faster than the state-dependent SDRE feedback control. However, as it does not update the states in each step, the OLFC control is more sensitive to parametric errors in the feedforward control than the feedforward control used in the SDRE control, whose results are also observed in [

The data can be requested from the corresponding author.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors acknowledge support by CNPq, CAPES, FAPESP, and FA, which are all Brazilian Research Funding Agencies.