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Nonlinear Dynamics and Chaos of Microcantilever-Based TM-AFMs with Squeeze Film Damping Effects.

Zhang WM, Meng G, Zhou JB, Chen JY - Sensors (Basel) (2009)

Bottom Line: The microcantilever was modeled as a spring-mass-damping system and the interaction between the tip and the sample was described by the Lennard-Jones (LJ) potential.The fundamental frequency and quality factor were calculated from the transient oscillations of the microcantilever vibrating in air.Results indicated the occurrence of periodic and chaotic motions and provided a comprehensive understanding of the hydrodynamic loading of microcantilevers.

View Article: PubMed Central - PubMed

Affiliation: State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, China; E-Mails: gmeng@sjtu.edu.cn (G.M.); giantbean@sjtu.edu.cn (J.-B.Z.); jerrysmiling@hotmail.com (J.-Y.C.).

ABSTRACT
In Atomic force microscope (AFM) examination of a vibrating microcantilever, the nonlinear tip-sample interaction would greatly influence the dynamics of the cantilever. In this paper, the nonlinear dynamics and chaos of a tip-sample dynamic system being run in the tapping mode (TM) were investigated by considering the effects of hydrodynamic loading and squeeze film damping. The microcantilever was modeled as a spring-mass-damping system and the interaction between the tip and the sample was described by the Lennard-Jones (LJ) potential. The fundamental frequency and quality factor were calculated from the transient oscillations of the microcantilever vibrating in air. Numerical simulations were carried out to study the coupled nonlinear dynamic system using the bifurcation diagram, Poincaré maps, largest Lyapunov exponent, phase portraits and time histories. Results indicated the occurrence of periodic and chaotic motions and provided a comprehensive understanding of the hydrodynamic loading of microcantilevers. It was demonstrated that the coupled dynamic system will experience complex nonlinear oscillation as the system parameters change and the effect of squeeze film damping is not negligible on the micro-scale.

No MeSH data available.


Related in: MedlinePlus

Bifurcation diagram of the cubic stiffness ratio β at different combinations of squeeze film damping ratios, material parameters and equilibrium parameters: (a) η = 0.08, Σ = 0.3, α = 1.2; (b) η = 0.08, Σ = 0.5, α = 1.2; (c) η = 0.14, Σ = 0.5, α = 1.2; (d) η = 0.14, Σ = 0.5, α = 1.6.
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f12-sensors-09-03854: Bifurcation diagram of the cubic stiffness ratio β at different combinations of squeeze film damping ratios, material parameters and equilibrium parameters: (a) η = 0.08, Σ = 0.3, α = 1.2; (b) η = 0.08, Σ = 0.5, α = 1.2; (c) η = 0.14, Σ = 0.5, α = 1.2; (d) η = 0.14, Σ = 0.5, α = 1.6.

Mentions: Figure 12 is the bifurcation diagram of the cubic stiffness ratio β on the response of TM-AFM tip-sample system at the interval of 0.3 < β < 0.6 for various combinations of squeeze film damping ratios, material and equilibrium parameters, and the bifurcation step Δβ = 0.001. It can be observed from Figure 12(a) that the response of the coupled system has a complete process from chaotic motion through periodic motion and chaotic motion to period-1 motion. At the interval of 0.3 < β < 0.43, the system response enters periodic motion from chaotic motion, then it becomes chaotic motion again, and finally it comes into steady-state motion with period-1 in the range of 0.43 < β < 0.6. With the increase of squeeze film damping η (η = 0.14), the system response changes noticeably and it mainly contains the periodic components, such as period-1, period-3 and period-6 motions, as illustrated in Figure 12(c). As the equilibrium parameter α increases, the chaotic components of the system response decrease, while the periodic components increase and contain period-2, period-4 and period-8 motions with the case of α = 1.6, as shown in Figure 12(d).


Nonlinear Dynamics and Chaos of Microcantilever-Based TM-AFMs with Squeeze Film Damping Effects.

Zhang WM, Meng G, Zhou JB, Chen JY - Sensors (Basel) (2009)

Bifurcation diagram of the cubic stiffness ratio β at different combinations of squeeze film damping ratios, material parameters and equilibrium parameters: (a) η = 0.08, Σ = 0.3, α = 1.2; (b) η = 0.08, Σ = 0.5, α = 1.2; (c) η = 0.14, Σ = 0.5, α = 1.2; (d) η = 0.14, Σ = 0.5, α = 1.6.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3297151&req=5

f12-sensors-09-03854: Bifurcation diagram of the cubic stiffness ratio β at different combinations of squeeze film damping ratios, material parameters and equilibrium parameters: (a) η = 0.08, Σ = 0.3, α = 1.2; (b) η = 0.08, Σ = 0.5, α = 1.2; (c) η = 0.14, Σ = 0.5, α = 1.2; (d) η = 0.14, Σ = 0.5, α = 1.6.
Mentions: Figure 12 is the bifurcation diagram of the cubic stiffness ratio β on the response of TM-AFM tip-sample system at the interval of 0.3 < β < 0.6 for various combinations of squeeze film damping ratios, material and equilibrium parameters, and the bifurcation step Δβ = 0.001. It can be observed from Figure 12(a) that the response of the coupled system has a complete process from chaotic motion through periodic motion and chaotic motion to period-1 motion. At the interval of 0.3 < β < 0.43, the system response enters periodic motion from chaotic motion, then it becomes chaotic motion again, and finally it comes into steady-state motion with period-1 in the range of 0.43 < β < 0.6. With the increase of squeeze film damping η (η = 0.14), the system response changes noticeably and it mainly contains the periodic components, such as period-1, period-3 and period-6 motions, as illustrated in Figure 12(c). As the equilibrium parameter α increases, the chaotic components of the system response decrease, while the periodic components increase and contain period-2, period-4 and period-8 motions with the case of α = 1.6, as shown in Figure 12(d).

Bottom Line: The microcantilever was modeled as a spring-mass-damping system and the interaction between the tip and the sample was described by the Lennard-Jones (LJ) potential.The fundamental frequency and quality factor were calculated from the transient oscillations of the microcantilever vibrating in air.Results indicated the occurrence of periodic and chaotic motions and provided a comprehensive understanding of the hydrodynamic loading of microcantilevers.

View Article: PubMed Central - PubMed

Affiliation: State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, China; E-Mails: gmeng@sjtu.edu.cn (G.M.); giantbean@sjtu.edu.cn (J.-B.Z.); jerrysmiling@hotmail.com (J.-Y.C.).

ABSTRACT
In Atomic force microscope (AFM) examination of a vibrating microcantilever, the nonlinear tip-sample interaction would greatly influence the dynamics of the cantilever. In this paper, the nonlinear dynamics and chaos of a tip-sample dynamic system being run in the tapping mode (TM) were investigated by considering the effects of hydrodynamic loading and squeeze film damping. The microcantilever was modeled as a spring-mass-damping system and the interaction between the tip and the sample was described by the Lennard-Jones (LJ) potential. The fundamental frequency and quality factor were calculated from the transient oscillations of the microcantilever vibrating in air. Numerical simulations were carried out to study the coupled nonlinear dynamic system using the bifurcation diagram, Poincaré maps, largest Lyapunov exponent, phase portraits and time histories. Results indicated the occurrence of periodic and chaotic motions and provided a comprehensive understanding of the hydrodynamic loading of microcantilevers. It was demonstrated that the coupled dynamic system will experience complex nonlinear oscillation as the system parameters change and the effect of squeeze film damping is not negligible on the micro-scale.

No MeSH data available.


Related in: MedlinePlus