Eigenvalues and dynamic stiffness of picket-shaped cantilevers
Introduction
In many microelectromechanical (MEMS) devices, accurate knowledge of the resonance frequencies is critically important. In addition, knowledge of the spring constant of beam-shaped cantilevers is important in applications such as measuring chemical bond forces, surface adhesion, mechanical properties, imaging self-assembled monolayers and determining the imaging or tapping force [1], to name a few. The dynamic spring constants, as well as the entire dynamic response to external excitations, depend on the beam geometry and its tip mass. The case of a uniform cantilever of rectangular cross section has been extensively studied and analytical expressions describing its dynamic behavior have been developed. Picket-shaped microcantilevers, Fig. 1, are widely available as commercial probes designed to suit different applications. For these, and other non-uniform geometries, one has to use detailed numerical methods, such as finite element methods (FE), when accurate dynamic responses are needed. Approximating non-uniform cantilevers as uniform ones was made in many studies in the past, but this approach introduces high uncertainty of the results that are not acceptable in cases where high fidelity is of the essence.
The objective of this article is to provide details of the variation patterns of both the resonance frequencies and dynamic stiffness of picket microcantilevers with the variation of both taper-ratio and tip mass-ratio. The results are presented in closed form, which should help in design, analysis and optimization of dynamic characteristics of MEMS devices that utilize these cantilevers.
Next section presents the closed-form solutions of the resonance frequency and dynamic stiffness of picket-shaped cantilevers with tip masses. Section 3 presents applications of the closed form solutions in case studies related to characterization of those microcantilevers. Section 4 presents a new calibration procedure based on the present study.
Section snippets
Eigenvalues and dynamic stiffness of picket-shaped cantilevers
For the picket-shaped microcantilever shown in Fig. 1, the resonance frequency “fn” of the transverse vibration in the y direction, the taper ratio “τ” and the tip mass ratio “μ” are
n is the mode number, λn is the eigenvalue, L is the length, E is Young’s Modulus, I0 = b0 h3/12, b0 is the width of the rectangular cross section at the clamped end, h is the constant thickness, Lt is the length of the tapered segment, Mtip is the tip mass, ρ is the
Comparisons with experimental data
In this section, predictions of the present study are compared with experimental results. When material properties and cantilever dimensions are reported in an experimental study, resonance frequencies and stiffnesses are calculated here and compared with the measurements. Otherwise, the frequency ratios and stiffness ratios are calculated and compared with the experiment.
Calibration of picket cantilevers
A survey of dozens of publications showed that among commercial microcantilevers, reported experimental results for Olympus AC240 cantilevers were the largest in number. Therefore, this microcantilever was used in this analysis. As specified by the vendor (Olympus brochure http://probe.olympus-global.com/en/), the following nominal values of cantilever dimensions were used in the numerical simulation presented below. L, b and h are 240, 40 and 2.1 μm, respectively. The taper length Lt = 44 μm;
discussion and conclusion
To illustrate how the closed form expressions in Table 4, Table 5 could be useful in tuning cantilever design to achieve desired characteristics, reference is made to a recent study [31]. In that study, the authors tailored the cantilever such that the second bending mode of the cantilever shifts to the frequency of the 6th Fourier component of the tip-sample interaction force, i.e. f2/f1 = 6. Consequently, the phase of the motion of the first and the second modes is synchronized in a way that
Funding
None.
Declaration of Competing Interest
None.
Mohamed A. Mahmoud received his Master’s and Doctorate degrees from the University of Waterloo, Canada. He worked as a Faculty at the State University of New York at Buffalo, University of Bridgeport in Connecticut, and The College of Technological Studies in Kuwait. He retired in 2018.
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Cited by (1)
Mohamed A. Mahmoud received his Master’s and Doctorate degrees from the University of Waterloo, Canada. He worked as a Faculty at the State University of New York at Buffalo, University of Bridgeport in Connecticut, and The College of Technological Studies in Kuwait. He retired in 2018.