Effects of Tip Mass and Interaction Force on Nonlinear Behavior of Force Modulation FM-AFM Cantilever

2016 ◽  
Vol 33 (2) ◽  
pp. 257-268 ◽  
Author(s):  
K. E. Torkanpouri ◽  
H. Zohoor ◽  
M. H. Korayem
Keyword(s):  
Frequency Shift ◽  
Nonlinear Model ◽  
Equations Of Motion ◽  
Nonlinear Behavior ◽  
Interaction Force ◽  
Beam Model ◽  
Frequency Shifts ◽  
Excitation Mode ◽  
Force Modulation ◽  
Tip Mass

AbstractInfluences of the tip mass, excitation mode of Frequency Modulated Atomic Force Microscope (FM-AFM) on the resonance frequency shift in force modulation (FM) mode are studied. Governing equations of motion are determined based on Timoshenko beam model with concentrated end mass. Approach point and base amplitude are set such that the FM-AFM remains just in FM mode. Either the linearized and nonlinear Derjaguin-Muller-Toporov (DMT) model are investigated. Then frequency shifts are determined for various interaction force regimes. It is showed the effect of tip mass on frequency shift is significant even for small tips. Nonlinear model shows lower frequency shifts in comparison with linearized model. It is showed that the amplitude of response is increased by increasing the tip mass and order of base excitation. Deviation of frequency shift between linearized and nonlinear solution are studied. It is declared that the error between linearized and nonlinear model is complicated. A deviation index is used for explaining behavior of error while tip mass and excitation mode are changed. It is showed, this index predicts the trend of error in all excitation modes and force cases. Behavior of system is linearizing by increasing the order of excitation, generally.

Nonlinear Interaction Force Analysis of Microcantilevers Utilized in Atomic Force Microscopy

2009 ◽  
Author(s):  
Sohrab Eslami ◽  
Nader Jalili ◽  
Ali Passian ◽  
Laurene Tetard ◽  
Thomas Thundat
Keyword(s):  
Atomic Force Microscopy ◽  
Interaction Force ◽  
Beam Model ◽  
High Frequencies ◽  
Force Microscopy ◽  
Distributed Parameters ◽  
Atomic Force ◽  
General Force ◽  
Cartesian Coordinate ◽  
Tip Mass

This paper presents an Euler-Bernoulli microcantilever beam model utilized in non-contact Atomic Force Microscopy (AFM) systems. A distributed-parameters modeling is considered for such system. The motions of the microcantilever are studied in a general Cartesian coordinate with an excitation at the base such that beam end with a tip mass is subject to a general force. This general force comprising of two attractive and repulsive parts with high power terms is taken as the atomic intermolecular one which has a relation with the displacement between the tip mass and the surface such that the total general force will be in the form of an implicit nonlinear equation. It is most desired to observe the effects of the base excitation in high frequencies on the tip van der Waals interaction force. Hence, the general force will produce a peak in the FFT spectrum corresponding to the frequency of the base.

Validation of a Belt Model for Prediction of Hub Forces from a Rolling Tire4

2009 ◽  
Vol 37 (2) ◽  
pp. 62-102 ◽  
Author(s):  
C. Lecomte ◽  
W. R. Graham ◽  
D. J. O’Boy
Keyword(s):  
Internal Pressure ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Prediction Method ◽  
Analytical Models ◽  
Beam Model ◽  
Impedance Boundary ◽  
Bending Waves ◽  
Tire Rotation ◽  
Three Dimensional Models

Abstract An integrated model is under development which will be able to predict the interior noise due to the vibrations of a rolling tire structurally transmitted to the hub of a vehicle. Here, the tire belt model used as part of this prediction method is first briefly presented and discussed, and it is then compared to other models available in the literature. This component will be linked to the tread blocks through normal and tangential forces and to the sidewalls through impedance boundary conditions. The tire belt is modeled as an orthotropic cylindrical ring of negligible thickness with rotational effects, internal pressure, and prestresses included. The associated equations of motion are derived by a variational approach and are investigated for both unforced and forced motions. The model supports extensional and bending waves, which are believed to be the important features to correctly predict the hub forces in the midfrequency (50–500 Hz) range of interest. The predicted waves and forced responses of a benchmark structure are compared to the predictions of several alternative analytical models: two three dimensional models that can support multiple isotropic layers, one of these models include curvature and the other one is flat; a one-dimensional beam model which does not consider axial variations; and several shell models. Finally, the effects of internal pressure, prestress, curvature, and tire rotation on free waves are discussed.

On the Nonlinear Behavior of Rotating Shafts

1991 ◽  
Author(s):  
Tatu Leinonen
Keyword(s):  
Theoretical Model ◽  
General Theory ◽  
Nonlinear Model ◽  
Nonlinear Behavior ◽  
Rotating Shaft ◽  
Rotating Shafts ◽  
Bending Behaviour

Abstract This paper presents a nonlinear model to describe the bending behaviour of a rotating shaft, based on the general theory of a bending bar. Justification for this theoretical model has been provided by tests, the resulting curves more closely fitting observed results than those of other models.

Thermo-optical switches using coated microsphere resonators

2001 ◽  
Vol 694 ◽  
Author(s):  
C. Tapalian ◽  
J.-P. Laine ◽  
P. A. Lane
Keyword(s):  
Resonant Frequency ◽  
Frequency Shift ◽  
Conjugated Polymer ◽  
Optical Switching ◽  
Optical Switches ◽  
Whispering Gallery Modes ◽  
Pump Light ◽  
Silica Microsphere ◽  
Frequency Shifts ◽  
Whispering Gallery

AbstractWe report optical switching by a silica microsphere optical resonator coated by a conjugated polymer. Microspheres were fabricated by melting the tip of an optical fiber and coated by dipping in a 1 mg/ml toluene solution of poly(2,5-dioctyloxy-1,4-phenylenevinylene) (DOO-PPV). The resonator properties were characterized by evanescently coupling 1.55 µm light propagating along a stripline-pedestal anti-resonant reflecting optical waveguide into optical whispering gallery modes (WGMs). WGM linewidths less than 2 MHz were measured, corresponding to cavity Q > 108. WGM resonant frequency shifts as large as 3.2 GHz were observed when 405 nm pump light with a power density of ~100 mW/cm2 was incident on the microsphere. The time constant of the observed frequency shifts is approximately 0.165 seconds, leading us to attribute the frequency shift to thermo-optic effects. Such a system should be capable of thermo-optically switching at speeds on the order of 10 kHz.

scholarly journals Vibration Characteristics of Piezoelectric Microbeams Based on the Modified Couple Stress Theory

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
R. Ansari ◽  
M. A. Ashrafi ◽  
S. Hosseinzadeh
Keyword(s):  
Boundary Conditions ◽  
Couple Stress ◽  
Natural Frequencies ◽  
Couple Stress Theory ◽  
Equations Of Motion ◽  
Modified Couple Stress Theory ◽  
Beam Model ◽  
Governing Equations ◽  
Stress Theory ◽  
Modified Couple Stress

The vibration behavior of piezoelectric microbeams is studied on the basis of the modified couple stress theory. The governing equations of motion and boundary conditions for the Euler-Bernoulli and Timoshenko beam models are derived using Hamilton’s principle. By the exact solution of the governing equations, an expression for natural frequencies of microbeams with simply supported boundary conditions is obtained. Numerical results for both beam models are presented and the effects of piezoelectricity and length scale parameter are illustrated. It is found that the influences of piezoelectricity and size effects are more prominent when the length of microbeams decreases. A comparison between two beam models also reveals that the Euler-Bernoulli beam model tends to overestimate the natural frequencies of microbeams as compared to its Timoshenko counterpart.

Influence of Bistable Plunge Stiffness On Nonlinear Airfoil Flutter

2021 ◽  
Author(s):  
Renan F. Corrêa ◽  
Flávio D. Marques
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Mechanical Vibration ◽  
Nonlinear Behavior ◽  
Limit Cycle Oscillation ◽  
Restoring Force ◽  
Aeroelastic System ◽  
Technical Literature ◽  
Practical Applications

Abstract Aeroelastic systems have nonlinearities that provide a wide variety of complex dynamic behaviors. Nonlinear effects can be avoided in practical applications, as in instability suppression or desired, for instance, in the energy harvesting design. In the technical literature, there are surveys on nonlinear aeroelastic systems and the different manners they manifest. More recently, the bistable spring effect has been studied as an acceptable nonlinear behavior applied to mechanical vibration problems. The application of the bistable spring effect to aeroelastic problems is still not explored thoroughly. This paper contributes to analyzing the nonlinear dynamics of a typical airfoil section mounted on bistable spring support at plunging motion. The equations of motion are based on the typical aeroelastic section model with three degrees-of-freedom. Moreover, a hardening nonlinearity in pitch is also considered. A preliminary analysis of the bistable spring geometry’s influence in its restoring force and the elastic potential energy is performed. The response of the system is investigated for a set of geometrical configurations. It is possible to identify post-flutter motion regions, the so-called intrawell, and interwell. Results reveal that the transition between intrawell to interwell regions occurs smoothly, depending on the initial conditions. The bistable effect on the aeroelastic system can be advantageous in energy extraction problems due to the jump in oscillation amplitudes. Furthermore, the hardening effect in pitching motion reduces the limit cycle oscillation amplitudes and also delays the occurrence of the snap-through.

Free and Forced Vibration of a Kinked Cantilever Beam

1999 ◽  
Author(s):  
B. S. Reddy ◽  
K. R. Y. Simha ◽  
A. Ghosal
Keyword(s):  
Cantilever Beam ◽  
Mode Shape ◽  
Forced Vibration ◽  
Equations Of Motion ◽  
Element Formulation ◽  
Kink Angle ◽  
Order Polynomial ◽  
Relationship Of ◽  
Tip Mass ◽  
Beam Mode

Abstract In this paper we use the assumed modes method to derive an analytical model of a kinked cantilever beam of unit mass carrying a kink mass (mk) and a tip mass (mt). The model is used to study the free and forced vibration of such a beam. For the free vibration, we obtain the mode shape of the complete beam by solving an eight order polynomial whose coefficients are functions of the kink mass, kink angle and tip mass. A relationship of the form f(mk,mt,δ)=mk+mt(4+103cosδ+23cos2δ)=constant appears to give the same fundamental frequency for a given kink angle, δ, and different combinations of kink mass and tip mass. To derive the dynamic equations of motion, the complete kinked beam mode shape is used in a Lagrangian formulation. The equations of motion are numerically integrated with a torque applied at the base and the tip response for various kink angles are presented. The results match those obtained from a traditional finite element formulation.

Nonlinear Dynamic Modeling of Elastic Beam Fixed on a Moving Cart and Carrying Lumped Tip Mass Subjected to External Periodic Force

2009 ◽  
Author(s):  
Fadi A. Ghaith
Keyword(s):  
Linear Model ◽  
Equations Of Motion ◽  
Elastic Beam ◽  
Open Loop ◽  
Beam Deflection ◽  
Mode Shapes ◽  
Euler Beam ◽  
Periodic Force ◽  
Assumed Mode Method ◽  
Tip Mass

In the present work, a Bernoulli – Euler beam fixed on a moving cart and carrying lumped tip mass subjected to external periodic force is considered. Such a model could describe the motion of structures like forklift vehicles or ladder cars that carry heavy loads and military airplane wings with storage loads on their span. The nonlinear equations of motion which describe the global motion as well as the vibration motion were derived using Lagrangian approach under the inextensibility condition. In order to investigate the influence of the axial movement of the cart on the response of the system, unconstrained modal analysis has been carried out, and accurate mode shapes of the beam deflection were obtained. The assumed mode method was utilized for approximating the beam elastic deformation based on the single unconstrained mode shapes. Numerical simulation has been carried out to estimate the open-loop response of the nonlinear beam-mass-cart model as well as for the simplified linear model under the influence of the periodic excitation force. Also a comparison study between the responses of the linear and nonlinear models was established. It was shown that the maximum values of the beam tip deflection estimated from the nonlinear model are lower than the corresponding values obtained via the linear model, which reveals the importance of considering nonlinear hardening term in formulating the equations of motion for such system in order to come with more accurate and reliable model.

scholarly journals Utilization of 2:1 Internal Resonance in Microsystems

Micromachines ◽  
2018 ◽  
Vol 9 (9) ◽  
pp. 448 ◽  
Author(s):  
Navid Noori ◽  
Atabak Sarrafan ◽  
Farid Golnaraghi ◽  
Behraad Bahreyni
Keyword(s):  
Internal Resonance ◽  
Mode Coupling ◽  
Equations Of Motion ◽  
Low Frequency ◽  
Nonlinear Behavior ◽  
Excitation Amplitude ◽  
Beam Structure ◽  
Nonlinear Mode Coupling ◽  
T Beam ◽  
Low Frequency Mode

In this paper, the nonlinear mode coupling at 2:1 internal resonance has been studied both analytically and experimentally. A modified micro T-beam structure is proposed, and the equations of motion are developed using Lagrange’s energy method. A two-variable expansion perturbation method is used to describe the nonlinear behavior of the system. It is shown that in a microresonator with 2:1 internal resonance, the low-frequency mode is autoparametrically excited after the excitation amplitude reaches a certain threshold. The effect of damping on the performance of the system is also investigated.

An Adaptive Self-Sensing Strategy for Piezoelectrically-Actuated Microcantilevers Used in Ultra-Small Mass Sensing Applications

2006 ◽  
Author(s):  
Miheer Gurjar ◽  
Nader Jalili
Keyword(s):  
Adaptive Estimation ◽  
Equations Of Motion ◽  
Optimization Method ◽  
The Self ◽  
Optimization Approach ◽  
Mass Sensing ◽  
Piezoelectric Patch ◽  
Sensing Applications ◽  
Tip Mass ◽  
Microcantilever Beam

This paper presents a mathematical model of a self-sensing microcantilever beam for mass sensing applications. Equations of motion are derived for a microcantilever beam with a tip mass and a piezoelectric patch actuator deposited on the cantilever surface. In the self-sensing mode, the same piezoelectric patch is used for actuation and sensing. Selfinduced voltage signals, which are extracted using a capacitive bridge mechanism, reveal frequency information of the vibrating beam, which in turn, reveals the particle mass. Equations of motion are obtained using the extended Hamilton's principle by considering the microcantilever as a distributed- parameters system. Two methods to estimate the unknown tip mass are presented. The first one is based on an inverse solution to the characteristic equation problem, while the second method uses a constraint-based optimization approach to estimate the tip mass. To improve the self-sensing performance, the need for adaptive estimation of the piezoelectric capacitance is stressed and an online estimation mechanism is presented. Simulations are presented to demonstrate the ability of the model to detect tip mass up to 0.1 femtogram (1 femtogram = 10-15 gm). Further simulation results demonstrate the working of constraint optimization method and adaptive self-sensing mechanism.

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