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Microcantilever physical model and schematic diagram. (a) Microcantilever schematic diagram. (b) Physical model of the microcantilever.
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The tapping mode is one of the mostly employed techniques in atomic force microscopy due to its accurate imaging quality for a wide variety of surfaces. However, chaotic microcantilever motion impairs the obtention of accurate images from the sample surfaces. In order to investigate the problem the tapping mode atomic force microscope is modeled an...
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... the mid 1980s the atomic force microscope (AFM) is a powerful tool to surface investigation in Engineering, Physics, Biology, Medicine and Nanotechnology. 1–4 Additionally, since 1993, the tapping mode atomic force microscope (TM-AFM) is one of the mostly employed dynamic AFM techniques due to its accurate imaging quality for a wide variety of surfaces. 5–7 A typical AFM system ( Figure 1) consists of a microcantilever probe with a sharp tip assembled on a piezoelectric scanner (PZT). The tip–sample interaction forces deform the microcantilever shape bending it upwards or downwards depending on whether the resultant force is attractive or repulsive. The position of the microcantilever is determined by a photo- diode in which a laser beam, reflected from the microcantilever’s surface, incides. 4 Since the AFM is actually a force sensor, the microcantilever bending is measured and translated to the corresponding force signal. A comprehensive review of the force–distance theoretical background, as well as the great variety of measurements that can be performed by the AFM, can be found in Cappella and Dietler. 8 The dinamic AFM techniques can be classified as noncontact AFM, such as frequency modulation AFM (FM-AFM), or tapping mode AFM (TM-AFM). The TM-AFM or intermittent contact AFM is an amplitude modulation AFM (AM- AFM) technique. For noncontact AFM techniques the tip–sample interaction forces are governed by a potential interaction. On the other hand, in the TM- AFM technique the tip–sample interaction forces include both friction and attraction/repulsion potential interaction forces. The tip–sample interaction forces involve short and long range intermolecular forces. Additionally, microcantilevers have several distinct eigenmodes and the tip–sample interaction forces are highly nonlinear. 6,7,9–11 The interest for nonlinear dynamics in the AFM grew from observations of instabilities that occur at certain amplitudes of operation of the microcantilever oscillation. 12,13 This problem leads to the need of a new AFM generation with reduced imaging forces and the development of new methods to improve material contrast and sensitivity. It has been experimentally observed that under certain conditions of operation the TM-AFM presents chaotic behavior. 14 The existence of chaotic behavior is undesirable since this type of complex motion impairs the AFM measurements, resulting in inaccurate and low resolution topographic images. The per- fomance of the AFM can be improved by eliminating the possibility of chaotic motion, which can be done either by changing the AFM operating conditions to a region of the parameter space where regular motion is assured, 15 or by designing an active contoller that stabilizes the system on one of its unstable periodic orbits. 16 In the following section the mathematical model of the TM-AFM is obtained, and the analysis of the dynamical system is performed considering: bifurcation diagrams, phase portraits, time histories, Poincare ́ maps, attraction basins, Lyapunov exponents, and simulations. Later the state feedback and time-delayed control techniques are implemented. Computer simulations are used in order to verify e ffi ciency and robustness to parametric errors of both control techniques. Finally, the concluding remarks are presented. In this section the mathematical model of the TM- AFM is developed. According to Jhang et al., 3 the first mode of vibration of the microcantilever can be considered as a mass-spring-damper system as shown in the lumped parameter model 17 in Figure 2. According to that, the microcantilever governing equation of motion is given ...
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Citations
... The results revealed that the chaotic motion of the system decreases with considering the squeeze film damping. This model developed in [19] and two control techniques (the optimal linear feedback control and the time-delayed feedback control) were used to suppress the chaotic motion in the tapping mode atomic force microscope. ...
Sensitivity and resolution of an atomic force microscope (AFM) can be affected by the nonlinear vibrational behavior of its microcantilever. So, the main purpose of this paper is to analyze the chaotic vibration of an AFM microcantilever under base excitation. Based on the modified couple stress theory (MCST) and the Euler-Bernoulli beam model, the nonlinear governing equations of motion are derived using the Hamilton's principle. Also, the squeeze film damping and the nonlinear microcantilever tip-sample surface interaction descripted by the Lennard-Jones potentials is considered. The governing partial differential equation is dis-cretized by means of the single-mode Galerkin's method and solved using the Runge-Kutta method. The effects of the initial distance between the microcantilever tip and sample surface, excitation amplitude and damping on the nonlinear dynamic response of the system are investigated by the bifurcation diagrams, maximum Lyapunov exponent, phase plane portrait, time history and Poincare´map. The results indicate that the vibrational behaviors of the microcantilever include 6T-periodic and chaotic motions. Based on the classical continuum mechanics theory, it is found that chaotic behavior initiates at much higher initial distances than predicted by MCST.
... where x, x , and x denote the displacement, velocity, and acceleration of the microcantilever beam in units of m, m/s, and m/s 2 , respectively. During motion, apart from external excitation fcosωt, the microcantilever beam system is also affected by van der Waals force FvdM, linear spring force, nonlinear spring force (compensating for the linear spring force, equivalent stiffness is expressed in k2), linear damping force, extruded film damping force (compensating for the linear damping force), and Casimir force FCas [21,[23][24][25]. z denotes the distance from the tip of the microcantilever beam to the surface of the sample to be measured, assuming that the distance between the tip samples when the microcantilever beam is at rest is l0, then 0 ( ) = + z l x t . ...
The tapping-mode atomic force microscope (TM-AFM) is widely used today; however, improper matching between the operating medium and the sampling time may lead to inaccurate measurement results. The relationship between the damping coefficient and the steady state of the TM-AFM microcantilever is investigated in this paper using multiple stability theory. Firstly, the effects of changes in dimensionless linear damping coefficients and dimensionless piezoelectric film damping coefficients on the motion stability of the system are examined using bifurcation diagrams, phase trajectories, and domains of attraction. Subsequently, the degrees of effect of the two damping coefficients on the stability of the system are compared. Finally, the bi-parametric bifurcation characteristics of the system under a specific number of iterative cycles are investigated using the bi-parametric bifurcation diagram in conjunction with the actual working conditions, and the boundary conditions for the transition of the system’s motion from an unstable state to a stable state are obtained. The results of the study show that to ensure the accuracy and reliability of the individual measurement data in 500 iteration cycles, the dimensionless linear damping coefficient must be greater than 0.01014. Our results will provide valuable references for TM-AFM measurement media selection, improving TM-AFM imaging quality, measurement accuracy and maneuverability, and TM-AFM troubleshooting.
... Through an intensive bibliographical research, it was observed that the OLFC has already shown to be efficient for problems considering numerical simulations, but it appears that there is a lack of work for its applications with experimental electronic systems applied to signal synchronism [17,18,[21][22][23]. ...
In this paper, an electronic circuit based on Macek equations was developed in which the phenomenon of hyperchaos was observed. Synchronization of signals was applied using the technique optimal linear feedback control (OLFC) and embedded in digital signal processor (DSP), with the circuit with hyperchaotic behavior as a master system and a periodic slave system embedded in the DSP along with the applied control technique. Within the context presented, it was possible to carry out the synchronization in phase between the systems, i.e., taking the system with periodic behavior to hyperchaos.
... These values are often contaminated with noise due to uncertainty in process and measurement. Considering the effect of noise and to improve the accuracy of the model, concept of parametric uncertainty [26,29,[46][47][48] is adopted here ...
Non-ideal high-speed rotors often exhibit the Sommerfeld effect characterized by nonlinear jumps and eventually gets destabilized. This paper presents a bifurcation analysis to attenuate the jumps in a non-ideal internally damped DC motor-driven shaft-disk system via magnetorheological (MR) fluid damper. A nonlinear hyperbolic tangent model of MR damper is proposed and linearized using equivalent linearization technique. To meet the demands of reliable engineering design, the system parameters of the proposed MR model are optimized using genetic algorithm (GA) with the consideration of parametric uncertainty. Then the system equations of the non-ideal rotor system are derived using Lagrangian formulation. Following, a characteristic equation of fifth-order polynomial in rotor speed is obtained through energy balance. The steady-state response is studied with the help of control current of the MR damper and subsequently verified through a bifurcation analysis with the help of root locus technique. A few results are also validated with earlier works. The nonlinear jumps are found to be attenuated as the control current of MR damper increases. The root locus technique confirms the jump phenomena though the existence of multiplicity of roots of the characteristic equation considering supply voltage as a gain. The proposed saddle-node bifurcation study confirms the cessation of the Sommerfeld effect when two unstable (saddle) points are found to be degenerated into a stable node at a specific bifurcation value of the MR control current. The nonlinear jumps of non-ideal rotor can be attenuated by altering the control current of the MR damper. This study also suggests that MR-based semi-active strategy is more effective than the AMB-based active control as the former takes much lesser current to attenuate the jumps. MR-based attenuation is found to be safer and more reliable than its active counterpart, i.e., AMB as the overall natural frequency of MR-based non-ideal rotor is far more behind the instability threshold of the system.
... In this section, the sensitivity of the three proposed controllers to parametric errors is investigated because in real process the parameters used in the control may contain parametric errors (Peruzzi et al. 2016;Balthazar et al. 2012;Avanço et al. 2018). To consider the effects of parametric errors uncertainties on the performance of the three proposed controllers, the parameters used in each control will be considered to have a random error of ± 20 % (Nozaki et al. 2013;Tusset et al. 2015). ...
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... 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 [11]. According to [12], it has been experimentally observed that the microcantilever of AFM in tapping mode may undergo chaotic behavior under certain conditions, due to the interactions of atomic scale forces between the surface and the tip of the AFM-TM [12]. e 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 [12]. ...
... 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 [11]. According to [12], it has been experimentally observed that the microcantilever of AFM in tapping mode may undergo chaotic behavior under certain conditions, due to the interactions of atomic scale forces between the surface and the tip of the AFM-TM [12]. e 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 [12]. ...
... According to [12], it has been experimentally observed that the microcantilever of AFM in tapping mode may undergo chaotic behavior under certain conditions, due to the interactions of atomic scale forces between the surface and the tip of the AFM-TM [12]. e 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 [12]. e authors provide the mathematical modeling of the dynamical equations of motion of the AFM-TM system to investigate the chaotic behavior of the microcantilever beam of the AFM-TM mechanism under VdW forces, where chaos is found and a control design has to be proposed. ...
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.
... Since a feedback controller of τ 2 is designed based on the mathematical model of the HSV. To analyze the influences of the parametric uncertainties on the performance of the excitation PWM controller, the parameters B v (damping coefficient of the ball valve) and R (equivalent resistance of the coil) used in the controller are considered to have a random error [34,35]. ...
High speed on/off valves (HSV) have often been used to control flow or pressure in digital hydraulic system due to higher switching frequency. However, the dynamic performance and energy efficiency are highly affected by the supply pressure and the carrier frequency. In this paper, a new adaptive PWM control method for HSV based on software is proposed. The proposed adaptive PWM consists of a reference PWM, an excitation PWM, a high frequency PWM, and a reverse PWM. Firstly, the nonlinear model of the HSV is established, and the structural composition and working principle of the proposed adaptive PWM control strategy are presented. Secondly, individual feedback controllers for the excitation PWM, the high frequency PWM, and the reverse PWM are designed, respectively; and each of the individual feedback controllers is experimentally verified. Finally, the comparative experimental results demonstrated that, with the proposed adaptive PWM control, the rising delay time of the control pressure drastically reduces by 84.6% (from 13ms to 2ms), the duty cycle’s effective range remains large (12%-85%) even with high carrier frequency (100 Hz), and that the temperature rise of the valve’s coil shell is reduced by 61.5%, compared to the three-voltage control. In addition, the dynamic performance and the energy efficiency of the HSV are not affected by the supply pressures and carrier frequencies, which proves that the proposed control strategy can improve the robustness and stability of the valve system.
... When compared to SEM and TEM, it is very practical that a sample preparation process is not as there is no need to cover the sample required in AFM (Colton et al. 1997;Kuznetsov et al. 2001;Kuznetsov and Mcphearson 2011). A schematic diagram of AFM is illustrated in Figure 5 ( Balthazar et al., 2013). within their physiological environment with the help of AFM. ...
Viruses are very small physical particles that can not be observed with a normal microscope. Only the largest virus, the poxvirus can be seen in the light microscope. Other tools are required for a detailed examination of viruses in ultrastructural size. Especially for the last 20 years, these advanced and ongoing tools have been used in the investigation of the biological molecules and biological processes of viruses. With the help of virus imaging tools, viruses can be identified in clinical samples, thereby explaining the detailed life cycle of the virus. Thus, these tools help the creation of new and safe vaccines and antiviral medicine. In this review, electron microscopy (EM) tools were given as scanning electron microscopy (SEM), transmission electron microscopy (TEM), and cryo-electron microscopy (Cryo-EM) since generally used tools based on EM. Besides EM-based tools, X-ray crystallography which is the basis for cryo-EM and atomic force microscopy (AFM) that is relatively economic and easy to apply compared to other microscopies were also described.
... In [16], the time-delayed feedback control is used in an Atomic Force Microscope microcantilever with chaotic behavior. The control of the chaotic behavior of the system was implemented by using the time-delayed feedback control. ...
... = 0.5, = 25, = 6.28, = 2, = 10, 10 = 1, and 20 = 1.5. The parameters 1 , 3 , , and were chosen in [16] such as ( 1 ≪ 2 0 0 ). ...
... where is the time delay and the feedback gain. The terms [ 1 ( − ), 2 ( − )] and [ 1 ( ), 2 ( )] represent scalar output signals measured at the current time and at the previous time ( − ), respectively [16,17,[24][25][26]. Since the control input (13) only depends on the output signal, the time delay is adjusted to the period of a target unstable periodic orbit that is intended to be stabilized in a chaotic attractor, and the control input therefore converges to null after the controlled system is stabilized to the target orbit. ...
The dynamic analysis and control of a nonlinear MEM resonator system are considered. Phase diagram, bifurcation diagram, and the 0-1 test are applied to the analysis of the influence of the parameters on the dynamics of the system, whose parameters are damping coefficient, polarization of the voltage, and nonlinear stiffness term. The bifurcation diagram is used to demonstrate the existence of the pull-in effect. Numerical results showed that the parameters, which were taken into account, were significant, indicating that the response can be either chaotic or periodic behavior. In order to bring the system from a chaotic state to a periodic orbit, two controls are considered: the time-delayed feedback control and the sliding mode control.
... These new concepts are aimed to improve the performance required in the analysis of nonlinear dynamical systems. In particular, problem of modeling and control for nonlinear dynamical systems showing chaotic behavior has attracted increasing attention because of its potential importance in actual real-world applications [5][6][7][8][9]. ...
In this paper, a fractional-order prediction-based feedback control scheme (Fo-PbFC) is proposed to stabilize the unstable equilibrium points and to synchronize the fractional-order chaotic systems (FoCS). The design of Fo-PbFC, derived and based on Lyapunov stabilization arguments and matrix measure, is theoretically rigorous and represents a powerful and simple approach to provide a reasonable trade-off between computational overhead, storage space, numerical accuracy and stability analysis in control and synchronization of a class of FoCS. Numerical simulations are also provided to verify the validity and the feasibility of the proposed scheme by considering the fractional-order Newton–Leipnik chaotic and the fractional-order Mathieu–Van Der Pol hyperchaotic systems as illustrative examples.
