Effect of tip mass on frequency response and sensitivity of AFM cantilever in liquid

In this work the effect of the application of an alternating magnetic field on the large transverse vibration of a cantilever beam with tip mass is investigated. The governing equation of motion is derived using D’Alembert’s principle, which is reduced to its nondimensional temporal form by using the generalized Galerkin method. The temporal equation of motion of the system contains nonlinearities of geometric and inertial types along with parametric excitation and nonlinear damping terms. Method of multiple scales is used to determine the instability region and frequency response curves of the system. The influences of the damping, tip mass, amplitude of magnetic field strength, permeability, and conductivity of the beam material on the frequency response curves are investigated. These perturbation results are found to be in good agreement with those obtained by numerically solving the temporal equation of motion and experimental results. This work will find extensive applications for controlling vibration in flexible structures using a magnetic field.

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DMCMN: Experimental/Analytical Evaluation of the Effect of Tip Mass on Atomic Force Microscope Cantilever Calibration

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.4000160 ◽

2009 ◽

Vol 131 (6) ◽

Cited By ~ 18

Author(s):

Matthew S. Allen ◽

Hartono Sumali ◽

Peter C. Penegor

Keyword(s):

Atomic Force Microscope ◽

Fundamental Mode ◽

Spring Constant ◽

Mode Shapes ◽

Contact Mode ◽

Modified Method ◽

Atomic Force ◽

Afm Cantilever ◽

Cantilever Probes ◽

Tip Mass

Quantitative studies of material properties and interfaces using the atomic force microscope (AFM) have important applications in engineering, biotechnology, and chemistry. Contrary to what the name suggests, the AFM actually measures the displacement of a microscale probe, so one must determine the stiffness of the probe to find the force exerted on a sample. Numerous methods have been proposed for determining the spring constant of AFM cantilever probes, yet most neglect the mass of the probe tip. This work explores the effect of the tip mass on AFM calibration using the method of Sader (1995, “Method for the Calibration of Atomic Force Microscope Cantilevers,” Rev. Sci. Instrum., 66, pp. 3789) and extends that method to account for a massive, rigid tip. One can use this modified method to estimate the spring constant of a cantilever from the measured natural frequency and Q-factor for any mode of the probe. This may be helpful when the fundamental mode is difficult to measure or to check for inaccuracies in the calibration obtained with the fundamental mode. The error analysis presented here shows that if the tip is not considered, then the error in the static stiffness is roughly of the same order as the ratio of the tip’s mass to the cantilever beam’s. The area density of the AFM probe is also misestimated if the tip mass is not accounted for, although the trends are different. The model presented here can be used to identify the mass of a probe tip from measurements of the natural frequencies of the probe. These concepts are applied to six low spring-constant, contact-mode AFM cantilevers, and the results suggest that some of the probes are well modeled by an Euler–Bernoulli beam with a constant cross section and a rigid tip, while others are not. One probe is examined in detail, using scanning electron microscopy to quantify the size of the tip and the thickness uniformity of the probe, and laser Doppler vibrometry is used to measure the first four mode shapes. The results suggest that this probe’s thickness is significantly nonuniform, so the models upon which dynamic calibration is based may not be appropriate for this probe.

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Torsional Sensitivity of the First Four Modes of an AFM Cantilever With a Sidewall Probe Using Analytical Method

Volume 6: ASME Power Transmission and Gearing Conference; 3rd International Conference on Micro- and Nanosystems; 11th International Conference on Advanced Vehicle and Tire Technologies ◽

10.1115/detc2009-87035 ◽

2009 ◽

Cited By ~ 2

Author(s):

M. H. Kahrobaiyan ◽

M. Rahaeifard ◽

M. T. Ahmadian

Keyword(s):

Resonant Frequency ◽

Analytical Method ◽

Contact Stiffness ◽

Vibration Modes ◽

Vertical Extension ◽

Cantilever Length ◽

Afm Cantilever ◽

Tip Mass ◽

Extension Length ◽

Very High

In this study, using analytical method, the torsional resonant frequency and torsional sensitivity of the first four modes of an AFM cantilever with sidewall probe including a horizontal cantilever and a vertical extension is analyzed and a closed form for torsional sensitivity of the probe is derived. In addition, the effect of relative parameters such as ratio of vertical extension length to horizontal cantilever length is investigated. According to this study, the results show that as contact stiffness increases, the resonant frequencies of all vibration modes increases until they reach constant values at very high values of contact stiffness. It is also can be found that low-order modes are more sensitive than high-order one. When contact stiffness increases, the torsional sensitivities of all vibration modes decrease and the graphs converge at very high values of contact stiffness. In addition, enhancement of ratio of vertical extension length to cantilever length decreases the resonant frequency of mode 1 for all values of the contact stiffness and decreases torsional sensitivity for low values of the contact stiffness. But for high values of contact stiffness, there is a peak for torsional sensitivity. Finally the result shows that increase of the tip mass, decreases the torsional sensitivity with a light slope.

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On the Dynamics of Vibro-Impacting Tapered Cantilever With Tip Mass

Volume 4B: Dynamics, Vibration, and Control ◽

10.1115/imece2016-65617 ◽

2016 ◽

Author(s):

Vishal Vyas ◽

Prasanna Gandhi

Keyword(s):

Frequency Shift ◽

Frequency Response ◽

Amplitude Ratio ◽

Turbine Blades ◽

Peak Amplitude ◽

Mode Shapes ◽

Vibration Energy Harvesting ◽

Response Curves ◽

Tapered Beam ◽

Tip Mass

Vibro-impact situations in beams arise in several applications including turbine blades, vibration energy harvesting, etc This paper presents dynamic analysis of vibro-impacting tapered beam having tip mass and impacting against two stops. It is well known in the literature that typical frequency response of a uniform cantilever impacting against single stop shows resonance frequency shift along with hysteretic jump and drop phenomena. This paper investigates interesting perturbations in the frequency response curves on account of addition of another stop for a tapered counterpart. A piecewise linear modeling is used to consider spring-damper model for impact and tapered beam mode shapes are obtained using differential transform method. It is demonstrated both experimentally and theoretically that more taper in the beam in general increases the peak amplitude ratio and two stops play important role in enhancement of peak amplitude ratio while keeping the frequency shift the same.

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Nonlinear Dynamic Response of Hysteretic Wire Ropes: Modeling and Experiments

Volume 8: 30th Conference on Mechanical Vibration and Noise ◽

10.1115/detc2018-86418 ◽

2018 ◽

Author(s):

Andrea Arena ◽

Biagio Carboni ◽

Walter Lacarbonara

Keyword(s):

Dynamic Response ◽

Frequency Response ◽

Nonlinear Dynamic ◽

Wire Rope ◽

Hysteretic Behavior ◽

Response Curves ◽

Nonlinear Dynamic Response ◽

Wire Ropes ◽

Tip Mass ◽

Constitutive Parameters

The nonlinear dynamic response of short cables with a tip mass subject to base excitations and undergoing primary resonance is investigated via experimental tests and by employing an ad hoc nonlinear mechanical model. The considered cables are made of several strands of steel wires twisted into a helix forming composite ropes in a pattern known as ‘laid ropes’. Such short span ropes exhibit a hysteretic behavior due to the inter-wire frictional sliding. A nonlinear one-dimensional (1D) continuum model based on the geometrically exact Euler-Bernoulli beam theory is conveniently adapted to describe the cable dynamic response. The Bouc-Wen law of hysteresis is incorporated in the moment-curvature constitutive relationship to reproduce the hysteretic behavior of short steel wire ropes subject to flexural cycles. The frequency response curves show a pronounced softening nonlinearity induced by hysteresis and inertia nonlinearity as confirmed by the experimental data acquired on a wire rope with a tip mass excited at its base by a shaker. The experimental nonlinear resonance response will be exploited to identify the constitutive parameters of the wire rope that best fit the frequency response curves at various forcing amplitudes.

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