**8. Conclusions**

62 Nonlinearity, Bifurcation and Chaos – Theory and Applications

 

**7.2. Local stability and PLL design for FM-AFM** 

From equation 39, and considering the phase error

2011), corresponds to a Constant phase error

considering

errors, i.e.,

factor and

 ,

equation 43 have negative real parts if:

Considering the filter coefficients <sup>2</sup>

2 *G <sup>n</sup>*

Additionally, from the design parameters

In addition, considering *<sup>c</sup>*

 

 

() () () *At At z t d dd* 

<sup>100</sup> () () () () () 0 *o o o oc*

 

*d t ADC c o*

 

 

<sup>100</sup>*Gsen ccc* 1 0

that represents the phase erros between the microcantilever oscillation and the PLL. The PLL behavior analysis is conveniently perfomed considering the cylindric state space,

asymptocally stable equilibrium point of equation 42 (See Bueno et al., 2010 and Bueno et al.,

that represents the PLL linear response to a frequency shift (step) of amplitude . The local stability of equation 42 can be determined by the position of the poles of equation 43, or by the Routh-Hurwitz criterion (See Bueno et al., 2010 and Bueno et al., 2011; Ogata, 1993). Therefore, considering that the coefficients of the filter are all positive and real, the poles of

> 0 1 0

> > and 1 2 *<sup>n</sup>*

Equation 45 establishes a design criterion that assure the local stability of the PLL, i.e., for small phase and frequency steps the PLL synchronizes to the microcantilever oscillation.

> and

and can be determined in order to satisfy the requirements of performance and stability.

*G* 

 0 0 *<sup>n</sup>* 

*<sup>n</sup>* the natural frequency, then, from equation 44, results:

0 . For small phase erros it can be considered that *sen*

*t* , (42) can be rewritten as:

 

 *o c* 

 

> 

(38)

*rt A At AGC c* (40)

, results that:

 

100 0 *G* (43)

. (44)

, where

. (45)

*<sup>n</sup>* the loop gain *G* has a superior bound,

is the damping

. In that case, the synchronous state, corresponding to an

*t* (41)

(42)

and to null frequency and acceleration

  in (42).

 *t t t Gsen t t* (39)

> This chapter deals with emergent problems in the Engineering Science research, presenting study and research related to NEMS systems, specially microcantilevers with many modes of vibration, for which the tip-sample interaction forces are highly nonlinear, impairing the stability of the latent image, while the others modes of vibration can be explored in order to improve the AFM performance.

In the context of this work, the following specific problem have been approached: The understanding of the relations of the properties and the structure of the nanoscopic and molecular materials, through the atomic force microscopy, using microcantilevers, giving subsidiary information to next generation of microscopy instrumentation.

On an Overview of Nonlinear and Chaotic Behavior and

Their Controls of an Atomic Force Microscopy (AFM) Vibrating Problem 65

[11] Beeby, S.P., Tudor, M.J., White, N.M., "Energy harvesting vibration sources for microsystems applications", Measurement Science and Technology 17, 2006, pp 175-195

[13] Bowen, W.R.; Hilal, N., Atomic Force Microscopy in Process Engineering - An

[15] Bueno, A.M.; Ferreira, A. A.; Piqueira, J.R.C.; Modeling and Filtering Double-Frequency Jitter in One-Way Master Slave Chain Networks. Circuits and Systems I: Regular

[16] Bueno, A. M.; Balthazar, J. M.; Piqueira, J. R. C. Phase-Locked Loop design applied to frequency-modulated atomic force microscope. Communications in Nonlinear Science

[17] Bueno, A.M.; Ferreira, A. A.; Piqueira, J.R.C.; Modeling and measuring. Communications In Nonlinear Science and Numerical Simulation 14 1854-1860, 2009. [18] Cidade, G. A. G.; Silva Neto, A. e Roberty, N. C Restauração de Imagens com Aplicações em Biologia e Engenharia: Problemas Inversos em Nanociência e Nanotecnologia Notas em Matemática Aplicada; 3 - Sao Carlos, SP : SBMAC, 2003, xiv,

[19] Chtiba, M. O., Choura, S., Nayfeh, A.H., El-Borgia, S. 2010.Vibration confinement and energy harvesting in flexible structures using collocated absorbers and piezoelectric

[20] Cottone, F., 2007, "Nonlinear Piezoelectric Generators for Vibration Energy Harvesting", Universita' Degli Studi Di Perugia, Dottorato Di Ricerca In Fisica, XX

[21] De Marqui Jr, C., Erturk, A., Inman, D. J. 2009. An electromechanical finite element model for piezoelectric energy harvester plates, Journal of Sound and Vibration 327

[22] Du Toit, N.E. and Wardle, B.L. 2007. ''Experimental Verification of Models for Microfabricated Piezoelectric Vibration Energy Harvesters,'' AIAA Journal, 45:1126—

[23] De Martini, B.E., Rhoads, J.F., Turner, K.L., Shaw, S.W., Moehlis, J., "Linear and Nonlinear Tuning of Parametrically Excited MEMS Oscillators" in Journal of

[24] Erturk, A., Inman, D.J., "On mechanical modeling of cantilevered piezoelectric vibration energy harvesters", Journal of Intelligent Material Systems and Structures, 2008 [25] Halliday, D., Resnick, R., Walker, J., Fundamentals of Physics, v. 3, 7th ed, Jonh Wiley

[26] Felix, Jl.P. and Balthazar, J.M, Comments on a nonlinear and nonideal electromechanical damping vibration absorber, Sommerfeld effect and energy transfer. Nonlinear Dynamics, (2009) Volume 55, Numbers 1-2, 1-11, DOI: 10.1007/s11071-008-

[27] Farrokh, A., Fathipour, M., Yazdanpanah, M.J., High precision imaging for noncontact mode atomic force microscope using an adaptive nonlinear observer and output state

[12] Bishop, R. H. The Mechatronics Handbook, CRC Press, 2002, ISBN: 0849300665.

Introduction to AFM for Improved Processes and Products, Elsevier, 2009.

[14] Bhushan, B; Handbook of Nanotechnology, Springer, Berlin, 2004.

devices, Journal of Sound and Vibration 329 (2010) 261–276.

Microelectromechanical Systems, vol. 16, no. 2, 2007, pp 310-318

Papers, IEEE Transactions on 57 3104 -3111, 2010.

and Numerical Simulation, (16)3835 – 3843, 2011.

88 p.

Ciclo.

1137.

(2009) 9–25.

and Sons, 2005.

9340-8
