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Eigenvalues and dynamic stiffness of picket-shaped cantilevers
Journal Pre-proof Eigenvalues and dynamic stiffness of picket-shaped cantilevers M.A. Mahmoud
PII:
S0924-4247(19)32265-4
DOI:
https://doi.org/10.1016/j.sna.2020.111872
Reference:
SNA 111872
To appear in:
Sensors and Actuators: A. Physical
Received Date:
15 December 2019
Revised Date:
20 January 2020
Accepted Date:
24 January 2020
Please cite this article as: Mahmoud MA, Eigenvalues and dynamic stiffness of picket-shaped cantilevers, Sensors and Actuators: A. Physical (2020), doi: https://doi.org/10.1016/j.sna.2020.111872
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Eigenvalues and dynamic stiffness of picket-shaped cantilevers M. A. Mahmoud Professor of Mechanical Engineering (Retired) 1222 Willow Point Dr, Murphy, Tx 75094, USA email: [email protected]
Highlights
Closed form expressions of the resonance frequency and dynamic stiffness are presented
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Graphical abstract
for picket-tapered cantilevers.
A new method for spring constant calibration is presented and verified.
Methods of application of the solution in sensing applications and design optimization are
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outlined.
Abstract A comprehensive data set is presented that covers eigenvalues and dynamic stiffness of picketshaped microcantilevers for a wide range of the taper ratio and tip mass ratios. The shape of these microcantilevers is such that most of the length is of uniform cross section and the end segment of 1
the beam is width-tapered to a point at the free end. This configuration is widely used in MEMS devices. Closed form expressions are presented for the eigenvalues, and for static and dynamic stiffness. These expressions could help designers and researchers in the MEMS field in general, and atomic force microscopy in particular.
Keywords: Spring constant; Dynamic stiffness; Tapered cantilever; MEMS; Atomic force
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microscopy; Calibration; Vibration; Resonance.
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1. Introduction
In many microelectromechanical (MEMS) devices, accurate knowledge of the resonance
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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,
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mechanical properties, imaging self-assembled monolayers and determining the imaging or tapping force (Gibson, 2005), to name a few. The dynamic spring constants, as well as the entire
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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
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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
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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.
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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. 2. Eigenvalues and dynamic stiffness of picket-shaped cantilevers For the picket-shaped microcantilever shown in Fig. 1, the resonance frequency “fn” of the
fn = λn2/(2π) √EI0 /(Mb L3 ),
= Lt /L,
and
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transverse vibration in the y direction, the taper ratio “” and the tip mass ratio “” are =Mtip /(Mb)
(1)
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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
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is the length of the tapered segment, Mtip is the tip mass, is the density, and Mb = b0 h L is the nominal cantilever mass. When = 0, the cantilever has a uniform cross section, and when =1
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the cantilever is fully tapered forming a triangular beam.
These two limiting cases were
investigated in (Mahmoud, 2020) in addition to trapezoidal-shaped beams.
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For partially tapered cantilevers, i.e. picket-shaped, there are no analytical solutions for the eigenvalues or the dynamic stiffness. Therefore, a numerical analysis based on the work of Myklestad (Mahmoud, 2019) was used in this study. In this analysis, the beam is discretized into
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a number of segments, the mass of each segment is lumped in the middle, and the beam is treated as a series of discrete masses connected by stiff, massless rods. This technique is also known in the literature as the Lumped Mass Transfer Matrix method. Details of how to best discretize nonuniform beams to ensure accurate results are given in (Mahmoud, 2019) where the accuracy of the
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Myklestad method’s predictions of the eigenvalues was validated for tapered beams with tip masses. The method was used in (Mahmoud, 2020) to determine closed form expressions of uniform and fully tapered microcantilevers with tip masses. For mode n, the dynamic stiffness ‘Kn’ (Melcher, 2007) and the normalized dynamic stiffness ‘n’ are 1
Kn = ∫0 EI() L [d2wn ()/d2 ]2 d / wn2 (1)
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and
n = Kn (L3/EI0)
(2)
wn() is beam deflection and EI() is the beam flexural rigidity, = x/L is the spatial coordinate along the beam (0 1). Eq. 2 is the result of equating expressions of the potential energy of the beam as a distributed mass system with the potential energy of the equivalent point mass system. Another dynamic property of interest is the equivalent mass, sometimes referred to as the effective mass. For mode n, the equivalent mass is the sum of the fraction (n) of beam mass that is equivalent to beam inertia effect plus the tip mass. It is denoted meq,n and is related to Kn by meq,n = (n + ) Mb;
n = λn4 (n + )
(3)
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Kn = 4 π2 fn2 meq,n ;
The dynamic stiffness Kn should not be confused with the static stiffness “Ks”. The static stiffness,
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Ks, is the vertical static load that acts at the cantilever free end and causes a unit tip deflection. It is also widely known as the spring constant, and is not affected by the presence of a mass attached
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at the free end. For the uniform cantilever, Ks = 3 EI0/L3. On the other hand, the dynamic stiffness ‘Kn’ is mode related and represents the gradient of the tip dynamic force with respect to the tip
is 6
EI0
3 +2 L3
and
s =
6 3 +2
(4)
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Ks =
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deflection for a particular resonance mode “n”. For a picket-shaped cantilever, the static stiffness
s is the normalized static stiffness. The derivation is presented in the appendix.
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The author has refrained from using the term “effective mass” to describe the quantity meq,n. It was found (Mahmoud, 2016) that many researchers equate the “effective mass” to 0.24Mb (or sometimes 0.236 Mb or ¼ Mb). meq,n identically equals ¼ Mb only for uniform tipless cantilevers. The equivalent mass meq,n is dependent on the geometry, tip mass and vibration mode.
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Tables 1 and 2 present results of ‘λn’ and ‘n’ for = 0 - 0.06.
Table 3 shows the results of this study together with numerical results of Melcher, 2007 and Hahner, 2010 for both a uniform cantilever and a tipless picket cantilever of taper ratio = 25%. For the uniform cantilever, exact results obtained using the closed form solution in (Mahmoud, 2020) are included. As evident in the table, the present results coincide with the exact results for
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the uniform cantilever, and the present results are in good agreement with the published results for the picket cantilever.
Further evidence of the validity and accuracy of the solution is presented in the next two sections. It is very useful for both the designer and the analyst to have closed form expressions of the eigen
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values and dynamic stiffness for each mode over the practical ranges of and . For this purpose, an expanded database was generated using the transfer matrix numerical analysis in the ranges 0
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0.15 and 0 0.5. The results in the database were curve fitted and the resulting expressions
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are presented in Tables 4 and 5.
Fig. 2 shows examples of the variation trends of the normalized resonance frequencies (λ12 and
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λ22) and the dynamic stiffnesses (1 and 2) due to variations of the taper ratio and tip mass. It can be seen that as the taper ratio increases, the resonance frequencies increase, but the frequency
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ratios decrease. Presence of a tip mass decreases the frequencies and their ratios. Two opposing trends are observed for the dynamic stiffness. For the 1st mode: as increases, 1 increases to a maximum then decreases. The presence of a tip mass decreases 1. For the higher modes: n and
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n/1 decrease as increases and the presence of a tip mass leads to increases of both n and n/1.
3. Comparisons with experimental data
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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. 3.1 Frequency shift due to an attached mass at an arbitrary location Fig. 3 presents the shift of the fundamental resonance frequency, f1, due to the presence of a micro sphere (mass=6.5x10-12 kg) at a variable distance Lm from the fixed end of a picket 5
microcantilever with a taper ratio = 12% (λ1=1.992 and s =2.997). Given f1=38.75 kHz and Ks =20.15 N/m (Xie, 2008), and combining Eqs. 1 and 4, then Mb = (λ14 Ks)/(4π2 s f12) =1.784x10-9 kg (i.e. =3.64x10-3). Other variables are given in the caption of Fig. 3. Using the transfer matrix analysis, the eigen value with the mass attached “λ1,” was calculated for a number of micro sphere locations, and the frequency shift was calculated from f1= f1 {(λ1, / λ1)2 -1}. The results are shown as a solid line in Fig. 3 and are seen to be in good agreement with the experimental measurements
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3.2 Resonance frequency and stiffness ratios of tipless cantilevers
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reported by Xie, 2008.
A tipless picket cantilever was used for mass sensing (Lakshmoji, 2012) and its first six resonance frequencies are shown in the first row of Table 6. The taper ratio was unknown. Using the present
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analysis, the taper ratio =0.075 gives the best agreement with the frequency ratios fn/f1 as shown in the table. The largest difference between the present analysis and the experiment is 0.37%,
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indicating good agreement.
Another tipless microcantilever was analyzed next. It is a commercial probe, model FORTA-TL by Applied Nanostructures. From a photograph of the microcantilever, the taper ratio, , was
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determined to be 0.05. For the first 2 modes, the measured dynamic properties fn , Knd and Qn (Lozano, 2010) are shown in Table 7. Using = 0.05, calculated λn and n are shown in the table together with the calculated ratios f2/f1 and K2/K1. There is excellent agreement between the
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calculated and measured ratios.
3.3 Frequency and stiffness of picket cantilever with probing tips A picket-shaped silicon cantilever with a trapezoidal cross section was studied in detail in (Mendels, 2006), both experimentally and using finite elements (FE). The taper ratio 0.06, the tip mass ratio was unknown, and the dimensions and material properties are given in the caption of Table 8. The combination =0.0606 and =0.01 gives the best agreement with the experiment. Tables 4 and 5 were used for modes 1-4 and the transfer matrix analysis was used for modes 5 and 6
6. The table shows good agreement between the present analysis and both the experimental and FE results, especially for the lower modes. In (Mendels, 2006), the FE analysis that was based on a fully-clamed end of the beam did not agree well with the experiment. The FE analysis agreed with the experiment only after a segment of the wafer was included in the model to account for support flexibility. The present analysis is based on a clamped end, and hence the slight deviation of the present results from the experimental and FE results for the higher modes. From Eq. 4, Ks= 0.0834 N/m, which is in excellent agreement with the measured value Ks=0.0836
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N/m in (Mendels, 2006).
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4. Calibration of picket cantilevers
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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 As specified by the vendor (Olympus brochure
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microcantilever was used in this analysis.
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 m, respectively.
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The taper length Lt= 44 m; =18.3%. The tip mass ratio was selected as 0.02. =2330 kg/m3. E=190 GPa. It should be emphasized that the objective here is observing the qualitative trends of
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the results and, therefore, the particular cantilever dimensions and properties used in the numerical simulation do not significantly alter these trends. A dataset was generated by randomly changing all variables within 2% of their nominal values, and subsequently using the expressions of λ1 and 1 from tables 4 and 5 to calculate f1 and K1 for each combination of the variables.
The
equivalent mass parameter 1 was calculated by rearranging Eq. 3 in the form
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2 K1 = 1 f1,kHz ;
1= 4 π2 meq,1 x106
(5)
f1,kHz is the 1st mode resonance in kHz and meq,1 is the equivalent mass of mode 1. Fig. 4 presents the simulation results that show a linear trend of K1 vs f12 whereas the equivalent mass parameter 1 does not vary in any specific way with respect to either K1 or f1. These results suggest that small variations of dimensions and material properties lead to small, random variations in the equivalent mass. This observation offers a means of cantilever calibration: when 1 is determined 7
experimentally for a set of similar cantilevers, K1 may be calculated from Eq. 5 for other similar cantilevers.
To test the hypothesis that the equivalent mass parameter 1 could be considered almost constant (i.e. changing within a small scatter band) for nominally similar microcantilevers, experimental results for 26 individual Olympus AC240 cantilevers (tested in air) reported in 13 separate publications were compiled and used in the analysis. Table 9 presents experimental results of 16
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cantilevers. K1 is plotted in Fig. 5a (solid circles) against f12 for each of these cantilevers. Results of 10 more cantilevers are plotted in the figure as triangles. Each triangular symbol represents the
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averges of 13 results of each microcantilever (obtained by different researchers) as reported by Sader et al., 2016. The experimental variation of K1 vs f12 in Fig. 5a suggests a linear trend whereas
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1 does not vary in any specific way with respect to f1 as shown in Fig. 5b.
From the slope of the best line fit, Fig. 5a, it is concluded that for Olympus AC240 microcantilevers
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tested in air, 1 = 3.56 x10-4 1.79 x10-5. Predictions of K1 were obtained using this value of 1 in Eq. 5 and the results are presented in Table 9. For completeness, the Sader Method (Sader et
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al. 2016) was also used to obtain estimates of K1 as shown in the last column of the table. Table 9 shows that the results of Eq. 5 are in reasonable agreement with the experiments and in some cases are in better agreement with experiments than the Sader method predictions. One
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principal advantage of Eq. 5 is that it only requires measuring f1. Sader’s method requires measuring both f1 and the quality factor Q1. However, the Sader method is more general because including Q1 accounts for effects of ambient fluid properties such as density and viscosity. Nevertheless, the results of Eq. 5 serve as a strong evidence of the validity of the expressions of λ1
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and 1 presented in Tables 4 and 5.
A significant advantage of Eq. 5 is that the calibration of the cantilever does not require tip-surface contact which is required for many methods of spring constant calibration. Tip-surface contact can be a disadvantage as it may cause changes in the tip shape of the AFM probe which will have significant effects on image resolution, measured adhesion and measured sample mechanical properties. Studies by Slattery et al, 2013 and Song et al, 2015 discuss how tip-surface contact 8
can change tip shape as well as various methods of spring constant and detector sensitivity calibration that do not require tip-surface contact. The Sader Method utilizes the formula K1 = C1 Q1 f11.3 , where C1 is a calibration coefficient. Both Eq. 5 and the Sader Method are based on the implicit assumption that effects of factors such as support compliance, undercuts and thickness variation are included within the scatter ranges of 1
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and C1.
One aspect not addressed in this study is the effect of imaging tip set-back observed in many
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commercial AFM cantilevers. Fortunately, the Olympus AC240 microcantilevers studied herein have their imaging tips right at their free ends. For other microcantilevers, there is a need for further investigation of the effect of tip set-back. This is important in view of the cubic dependence
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of Kn on the distance “L” between the base of the cantilever and the imaging tip, as in Eq. 2. This effect is particularly important for short cantilevers where the tip set-back could be significant
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relative to the length (Slattery et al, 2014, 2019). Theoretical work would be useful, but also experimental research is needed. Methods of cantilever calibration that can determine the spring
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constant at specific points or positions along the length of cantilevers of different geometries are required. The inverted reference cantilever method described by Slattery et al, 2013 may be useful for such experiments. Also, the added mass method could be used by changing the position of the
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added mass along the length of cantilever as proposed by Vakarelski et al, 2007. Until further research into the tip set-back effect is conducted, an approximate adjustment factor that accounts for this effect proposed by Sader et al., 2012 may be utilized. It should be noted that in addition to calibration of the cantilever spring constant, there are a
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number of other important calibrations that must be done to the entire AFM system for quantitative measurements and these include the cantilever detector sensitivity and the piezoelectric actuator. A paper by Wagner et al, 2011 discusses these calibrations and provides a good overview. 5. discussion and conclusion To illustrate how the closed form expressions in Tables 4 and 5 could be useful in tuning cantilever design to achieve desired characteristics, reference is made to a recent study (Keyvani, 2017). In 9
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 the speed of the tip is reduced right before touching the sample surface. Hence, the proposed cantilever indents the sample less and, therefore, applies a lower force. Their proposed cantilever was linearly tapered along the whole length such that the width at the free end is smaller than the
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width at the clamped end.
It is possible to achieve the same goal, i.e. f2/f1=6, using a picket-shaped cantilever. Fig. 6 is a
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plot of some of the present results and shows the correlations of , , f2/f1 and 2/1. Intersections of a vertical line through f2/f1= 6 with the grid of and curves yield a variety of and
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combinations to obtain f2/f1 =6. Some of these (, ) combinations are (0.2, 0.01), (0.2, 0.126), (0.25, 0), (0.17, 0.022) and (0.222, 0.0045). It is evident from the figure that the ratio f2/f1 = 6 is
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not attainable for 0.15.
The present analysis could also be useful in sensing applications. As an example, consider two
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cantilevers of identical dimensions and material properties differing only in that one is uniform and the other is picket-shaped with =0.25. When a particle of (normalized) mass adsorbs to the apex of the uniform cantilever, the sensitivity of mass sensing is Su ∝ (λ2n,after − λ2n,before )u /.
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When an identical particle adsorbs to the apex of the tapered beam, the sensitivity of mass sensing is Stapered ∝ (λ2n,after − λ2n,before )tapered /. The ratio of the average mass sensitivities between the two cantilevers is Stapered/Su =
(λ2n,after −λ2n,before )tapered (λ2n,after −λ2n,before )u
. As an example, for mode 2 and =0.001 and
using table 4, Stapered/Su = (5.18522-5.22)/(4.68942-4.6942) 3.6. For the 1nd and 3rd modes, the
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tapered beam’s sensitivity is higher than the corresponding uniform beam’s sensitivity by a factor of 2.1 and 5.2, respectively. In conclusion, closed form expressions of the resonance frequency and dynamic stiffness, based on results of numerical analysis, were presented for picket-shaped cantilevers with tip masses. These cantilevers are widely used in MEMS devices as they offer the possibility of tailoring the resonance frequency and the dynamic stiffness to suit different applications. Picket-shaped cantilever beams are softer (more compliant) but have larger resonance frequencies than uniform 10
beams. The resonance frequencies and stiffness vary in different patterns depending on the taper ratio, the tip mass and mode order. The closed form expressions presented in Tables 4 and 5 agree well with experiments.
Author statement
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I declare that I am the sole author. I declare that this research was not funded by any organization.
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M.A. Mahmoud
Appendix: Static spring constant of picket-shaped cantilevers
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A plan view of a picket-shaped cantilever is shown in Fig. A-1. The bending moment ‘Mz(x)’ at
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distance x due to a force ‘F’ normal to the plane of drawing acting at x=L is Mz(x) = F (L-x). Beam width, bx, in the tapered segment at distance x (see Fig. A-1) is bx = b0 (L-x) / L; EI(x) = EI0 (L-x) / L.
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In the range 0x L (the vertical deflection is denoted y1): y1`` (x) = F(L-x)/ EI0 ; y1` (x)= F (L x - x2/2) / EI0 ; y1(x)= F (L x2/2 – x3/6) / EI0
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In the range L x L (notice that + = 1 and the vertical deflection is denoted y2), y2`` (x) = F(L-x)/ EI(x) = F L / EI0 ; y2` (x) = F Lx / EI0 + C ; y1` ( L) = y2` ( L) yields C = ½F 2L2 / EI0 y1( L) = y2( L) yields D = – F 3L3 / 6EI0 y2(L)= F ( L 3/2) / EI0 + C L + D = {(3+2)/6}{FL 3/ EI0}
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y2(x)= ½F L x2 / EI0 + C x + D
Ks =
F y2 (L)
=
6
EI0
3 +2
L3
, and s =
6 3 +2
is the normalized static stiffness.
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Author Biography 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.
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of
He retired in 2018.
Fig. 1. Picket-shaped microcantilever.
-p
y L Lt
Mtip
lP
= Lt / L
re
x
Jo
ur na
z
14
Fig. 2. Variation trends of the normalized resonance frequencies and dynamic stiffness due to variations of taper ratio and tip mass.
λ2 2
4 0.1 0.2 0.3 0.4 0.5
3.2
1
3
s
25 20 180 0
3 0
f2/f1
5
6.5
2
130 80 30
2.8 0
Taper ratio
0.1 0.2 0.3 0.4 0.5
35 10
0
0.1 0.2 0.3 0.4 0.5
5.5 4.5 60 0
0.1 0.2 0.3 0.4 0.5
2/1
λ12
30
0.1 0.2 0.3 0.4 0.5
of
6
0
Taper ratio
0.1 0.2 0.3 0.4 0.5
Jo
ur na
lP
re
-p
ro
Taper ratio
15
Fig. 3. Frequency shift, f1, due to the presence of a micro sphere at Lm. The symbols are experimental results of Xie, 2008. Taper ratio =0.12. f1=38.75 kHz, L=598 m, b0=141m, added mass=6.5x10-12 kg. Static spring stiffness Ks=20.15 N/m and =3.64x10-3.
Analysis Series1
(Xie, 2008) Series2
0
-200
L
of
f1 , Hz
-100
Lm -400 0.2
0.4
0.6
0.8
1
-p
0
ro
-300
Jo
ur na
lP
re
Lm /L
16
Fig. 4. Simulation results. Variation of K1 vs f12 due to 2% random errors in all variables results in a linear trend of K1 vs f12 whereas 1 does not vary in any specific way with respect to K1 and f1. =0.02, =18.3%. 0.0004
1.6
1
1.4
0.0003 1.1
1.3
1.2
1.3
1.4
1.5
1.6
K1, N/m
1.2
0.0004
1.1
1
1 0.9 3000
0.0003 3500
4000 2
56
4500
58
of
K1, N/m
1.5
60
62
f1, kHz
2
Jo
ur na
lP
re
-p
ro
f1 , kHz
17
64
66
Fig. 5. (a) Experimental variation of K1 vs f12 for 26 Olympus AC240 microcantilevers. (b) The equivalent mass parameter 1 = K1/f12 does not vary in any specific way with respect to f1. 3.5
(a)
3
(b) 0.0006
2
1
1.5
0.0004 0.0002
1
2 K1 =0.000356 f1,kHz
0.5 0 3000
0 55
65
75
of
K1, N/m
2.5
f1, kHz
4000
5000
6000
7000
Jo
ur na
lP
re
-p
ro
f12, kHz2
18
85
Fig. 6. The resonance and stiffness ratios (modes 2 and 1) as functions of the taper ratio (continuous lines) and tip mass ratio (dotted and dashed lines). K2/K1 = 2/1.
80 70 = 0.15
50
0.12
40
0.10 0.08
30
of
K2 / K1
60
20
ro
10 0 5.25
5.5
5.75
6
6.25
6.5
Jo
ur na
lP
re
-p
f2 / f1
19
Fig. A-1. Plan view of a picket-shaped cantilever.
L x +=1
bx
b0
x L
Jo
ur na
lP
re
-p
ro
of
L
20
Table 1. Eigen values ‘λn’ and normalized dynamic stiffness ‘n’ for =0 and =0.02. =0
λ1
= 0.02 λ2
λ3
λ4
1
2
3
4
λ1
λ2
λ3
1
λ4
2
3
4
1.875 4.694 7.855 10.996
3.091 121.4 951.6 3654
1.839 4.610 7.722
10.820 3.077 138.4 1223 5298
0.051
1.923 4.810 8.044 11.253
3.111 104.0 714.1 2417
1.883 4.701 7.848
10.962 3.092 118.6 923.9 3579
0.101
1.972 4.922 8.212 11.462
3.131 90.64 560.2 1724
1.927 4.782 7.940
11.026 3.108 103.1 723.5 2591
0.149
2.020 5.021 8.347 11.607
3.151 80.39 453.7 1284
1.970 4.849 7.988
11.006 3.121 90.93 583.7 2017
0.2
2.072 5.117 8.459 11.700
3.172 71.21 365.7 957.0
2.015 4.906 7.995
10.920 3.133 79.92 474.0 1705
0.222
2.095 5.155 8.497 11.722
3.180 67.62 333.6 851.2
2.035 4.926 7.984
10.873 3.137 75.60 436.6 1638
0.25
2.123 5.200 8.535 11.737
3.189 63.42 298.2 744.9
2.060 4.947 7.962
10.819 3.140 70.61 399.8 1614
0.278
2.152 5.241 8.563 11.743
3.198 59.50 267.5 662.7
2.085 4.963 7.930
10.769 3.142 65.97 370.0 1617
0.303
2.178 5.274 8.581 11.744
3.205 56.15 243.6 604.4
2.107 4.973 7.898
10.733 3.142 62.11 349.8 1646
0.333
2.210 5.310 8.594 11.744
3.211 52.40 219.4 549.4
2.134 4.980 7.858
0.379
2.257 5.355 8.602 11.746
3.215 47.30 190.8 486.7
2.173 4.980 7.798
0.422
2.302 5.387 8.603 11.752
3.213 43.05 170.4 439.7
2.209 4.972 7.749
0.5
2.381 5.425 8.602 11.760
3.191 36.68 144.1 371.2
2.272 4.940 7.680
ro
of
0
10.699 3.140 57.90 332.8 1702 10.657 3.130 52.37 316.0 1780
-p
10.628 3.114 48.04 309.3 1863 10.573 3.063 42.06 306.6 2000
= 0.4
= 0.06
λ2
λ3
λ4
1
2
3
4
λ1
λ2
λ3
λ4
1
2
3
4
1.807 4.542 7.625 10.704
3.066 157.3 1551 7428
1.776 4.486 7.552 10.624 3.058 178.1 1932 10026
0.051
1.846 4.614 7.713 10.787
3.078 135.4 1199 5304
1.813 4.545 7.615 10.674 3.067 154.3 1531 7539
0.101
1.886 4.675 7.761 10.790
3.090 118.3 965.5 4156
1.850 4.591 7.639 10.651 3.076 135.8 1272 6315
0.149
1.925 4.721 7.767 10.724
3.098 104.9 808.9 3613
1.884 4.622 7.624 10.575 3.081 121.4 1106 5922
0.2
1.965 4.754 7.732 10.621
3.105 92.85 695.9 3513
1.920 4.640 7.574 10.481 3.084 108.6 999
6218
0.222
1.983 4.763 7.706 10.578
3.106 88.17 660.6 3580
1.936 4.643 7.545 10.445 3.083 103.7 970
6518
0.25
2.005 4.770 7.670 10.536
3.106 82.87 630.8 3770
1.955 4.642 7.507 10.413 3.081 98.25 953
7069
0.278
2.026 4.772 7.629 10.500
3.104 78.00 609.8 3999
1.974 4.637 7.467 10.388 3.077 93.28 947
7677
0.303
2.045 4.770 7.592 10.478
3.100 74.03 600.3 4252
1.991 4.629 7.434 10.374 3.071 89.31 955
8300
0.333
2.068 4.764 7.552 10.461
3.094 69.82 599.0 4586
2.011 4.617 7.399 10.366 3.062 85.21 979
9092
0.379
2.101 4.746 7.498 10.439
3.077 64.43 608.1 5044
2.039 4.591 7.355 10.354 3.041 80.10 1031 10177
0.422
2.131 4.723 7.460 10.424
3.053 60.45 629.9 5474
2.065 4.564 7.327 10.347 3.014 76.60 1099 11177
0.5
2.182 4.674 7.411 10.392
2.990 55.41 680.6 6183
2.105 4.513 7.293 10.325 2.945 72.79 1236 12839
ur na
0
Jo
λ1
lP
re
Table 2. Eigen values ‘λn’ and normalized dynamic stiffness ‘n’ for =0.04 and =0.06.
Table 3. Normalized dynamic stiffness ‘n’ and equivalent mass ‘βn’ for =0. =0 n
βn
=0.25 n
21
βn
βn,a *
n
Present Exact Melcher† Hahner‡ Present Exact Melcher Hahner Present Melcher Hahner Present Present Melcher Hahner
1
3.096
2 3
3.096 3.09
3.09
0.25
0.25 0.249
0.25
3.19
3.21
3.21
0.157
0.179 0.178
0.18
121.38 121.38 120.6
120.6
0.25
0.25 0.248
0.248
63.41
63
63.9
0.087
0.099 0.098
0.1
951.64 951.64 936
939
0.25
0.25 0.245
0.247
298.2
290.1
302.4
0.056
0.064 0.062
0.065
* βn,a is the effective mass normalized relative to the actual mass of the picket cantilever; βn = βn,a (1-/2). † Melcher et al., 2007. ‡ Hahner, 2010.
Table 4. Expressions for the eigenvalues ‘λn’ in the range 0 0.15 and 0 0.5. =10 . ψ =10 .
0.23%
0.12 %
Expression
of
0.16%
†
λ1 = (1.8751-1.85+4.12- 6.4 3) (0.0732 +0.5092 +1) {1- (2.4 2 + 1.8 ) +(18 3 + 8.5 2 + 5.9 ) 2 + (-55.5 3-17.32- 11) 3} λ2=4.694 -0.465 +0.247 2-0.094 3+0.017 4 +( 0.231-0.29 +0.3042-0.2433+0.114 4-0.0223 5) ψ +( -0.0013 -0.0183 0.062+ 0.09663-0.0414 4-0.00367 5+.00444 6) ψ2 +( -0.005 -0.0181 + 0.05272 -0.03593-0.0103 4+0.02057 50.00613 6) ψ3 +( 0.00038+ 0.002198 -0.002628 2 -0.003673+0.00895 4-0.0061246 5+0.00142 6) ψ4
ro
Accuracy*
(0.15 > > 0.085)∩(0.5 > > 0.22)
0.37%
-p
λ3= 9.88 2 - 5.11 + 8.372 +(1134 4 - 652.16 3 + 143.62 2- 14.2 + 0.135) ψ+(-3.19 2 + 0.963 - 0.001) ψ2+(-2.88 3 + 1.3033 2- 0.20763 + 0.0072) ψ3 Full range except (0.15 > >0.085)∩(0.5> >0.22)
0.21%
re
λ3=7.845-0.819+ 0.955 2-0.9 3 + 0.466 4-0.095 5+( 0.46-0.343 -1.078 2+ 2.162 3-1.39 4+ 0.303 5) ψ+( -0.0835-0.5354 + 1.761 2-2.189 3+ 1.1937 4 -0.239 5) ψ2 +( 0.00414+ 0.0694 -0.1717 2+ 0.1716 3-0.0714 4 + 0.00954 5) ψ3 <0.033
lP
λ4= χ {1+(-1 -1.624 ψ+1.116 ψ2-1.382 ψ3+0.4857 ψ4-0.0706 ψ5+0.003688 ψ6) +(11.6+29.1 ψ -7.12 ψ2-19.11 ψ3+23.95 ψ4-6.391 ψ5+0.5225 ψ6) 2+(-103.8+336.2ψ-2525 ψ2 +3830 ψ3-2094.3 ψ4+435.08 ψ5-31.33 ψ6) 3+(553.5-13215ψ+54487 ψ2-70977 ψ3+34586 ψ4-6750 ψ5+467 ψ6) 4} χ = -172.02 5 + 219.29 4 - 82.9 3 + 0.8149 2 + 5.1869 + 10.996 0.22%
0.15>>0.026
ur na
λ4= χ {1- 0.114 + 2.64 2-31.47 3+122.8 4-203.5 5+ 123.7 6+( -0.85 -5.29 -122.12+ 882.6 3 -1990.5 4+ 1493.3 5) + (7.09+ 24.4 + 2072.52-12884 3+27348 4-19761 5) 2+( -33.7-2.17 -13933 2+81260 3-167187 4+118236 5) 3+(66.9193.4 + 336152-189217 3+381833 4-266209 5) 4}
* Largest percent difference between fitted and numerical results. † The accuracy is 0.16% in the range {0 0.1 and 0 0.42}, 0.22% for {0.1< 0.15 and 0 0.42}, and 0.4% for 0.5> > 0.42.
Table 5. Expressions for the normalized dynamic stiffness ‘n’ in the range 0 0.15 and 0 0.5. =10 . ψ =10 . = 100 . Accuracy
1 = (3.0904-0.73+3.62-83)+ (0.38-5.1+40.62-1923+3954) +(0.33-3.4+9.62) 2+ (-0.76-52+6602-36403+76304) 3 +(-1.27+39+192-5,6153+47,8904-125,0005) 4
Jo
0.07%
Expression
0.18%
<0.033 2 = (121.375+79.65 +28 2-6.92 3+3.7 4)+(-395-281 +162 2-159 3-56 4+213 5) +(1142+476 -419 2+707 3+247 4) 2 +(-2,818-3,208 +8,275 2-1,084 3-38,098 4+52,044 5) 3 +(4,010+8,663 -12,742 2-53,763 3+27,9831 4-499,532 5+339,519 6) 4 +(-2,276-6,931 +4,544 2+62,910 3-240,080 4+363,896 5-209,541 6) 5
0.32%
>0.032 2 =(121.286+80.5 +25 2-2.52 3+0.48 4)+(-397-259.4 +83.6 2-66 3+29.5 4-5.48 5) +(1173+190 +478.7 2-181.9 3+32.6 4) 2+(-3,067-922 +1,992 2-3,744 3+2,283 4-496 5) 3 +(4,200+7,874 -22,108 2+34,629 3-27,549 4+11,271 5-1,874 6) 4 +(-2,315-7,844 +20,542 2-29,813 3+22,877 4-9,150 5+1,498 6) 5
0.32%
<0.036
22
3 = (951.61+1208 +779 2-201 3+248 4-282 5+188.5 6)+(-594.17-776 +557 2-515.7 3-3599 4+12856 5-13015 6) ψ +(300.37+267.6 -420 2-1081 3+16260 4-46055 5+43624 6) ψ2 +(-123.053-203 +750 2+1297 3-18714 4+50829 546817 6) ψ3 +(31.84+91.2 -238.43 2-1305.6 3+10278 4-25005 5+21870 6) ψ4 +(-4.3482-17.177 +26.96 2+349.3 32164.8 4+4904.3 5-4124 6) ψ5 +(0.2383+1.146 -0.8457 2-28.71 3+156.27 4-336.8 5+274.6 6) ψ6 >0.035 3 = (948.478+1229.4 +700.8 2-39.3 3+8.06 4)+(-567.5-670.76 +74.03 2+1.486 3-16 4+5 5) ψ +(220.294+131 +213.82 2-123.83 3+32 4) ψ2 +(-72.08+84.52 -285 2+287.63 3-143.3 4+28.031 5) ψ3 +(13.93-32.856 +97.88 2-120.237 3 +86.9 4-33.61 5+5.39 6) ψ4 +(-0.993+2.532 -7.263 2+8.386 3-5.922 4+2.251 5-0.356 6) ψ5
0.59%
(<0.035)∩(>0.149) 4 =(2819.209-547.3 +1724.234 2-1373.32 3+486.53 4-86.611 5+6.218 6)+(-1056.1+2512.4 -3809 2+2986.95 3-1011.57 4+172.439 5-11.842 6) ψ +(-266.04-3008.1 +3529.61 2-2554.266 3+814.55 4-130.774 5+8.399 6) ψ2 +(290.016+1582.76 -1575.62 2+1074.497 3-321.332 4+47.8518 5-2.782445 6) ψ3 +(-80.893-422.11 +377.567 2244.237 3+68.729 4-9.4308 5+0.480932 6) ψ4 +(9.9374+56.172 -46.609 2+28.746 3-7.6299 4+0.9557 5-0.04042 6) ψ5 +(-0.465-2.9693 +2.3288 2-1.37537 3+0.34474 4-0.03888 5+0.001214 6) ψ6
0.59%
Full range except (<0.035)∩( >0.149) 4 = (3,653.433+6,945 +6,550 2-1,135 3+921 4-411 5+75.7 6)+(-3,281-5,385 -856 2+3,144 3-3,287 4+1,630 5-314 6) ψ +(2,083-449 +11,445 2-18,423 3+16,507 4-7,534 5+1,376 6) ψ2 +(-970.6+1,592 -6,757 2+9,251 3-6,332.6 4+2,113.4 5-262 6) ψ3 +(279.92-543.9 +1,805.5 2-1,274.45 3-76.7 4+523.2 5-174 6) ψ4 +(-42.42+72.75 -221.77 266.47 3+324.7 4-254.86 5+63.73 6) ψ5 +(2.556-3.43 +9.977 2+18.2 3-35.95 4+24.33 5-5.723 6) ψ6
ro
of
0.32%
1
2
3
Experimental fn, kHz
12.5945
78.471
219.36
6.231
17.417
Experimental fn/f1 Analytical λn
1.9467
4.8651
Analytical fn/f1 = λn / λ1
6.246
Difference
0.24%
2
5
6
429.272
708.084
1,056.398
34.084
56.222
83.878
8.1284
11.3606
14.5835
17.796
17.435
34.057
56.121
83.569
0.10%
-0.08%
-0.18%
-0.37%
lP
2
4
re
Mode n
-p
Table 6. Comparison of experimental (Lakshmoji, 2012) and analytical resonance ratios for a tipless, picket-shaped cantilever. =0.075.
ur na
Table 7. Comparison of experimental measurements in (Lozano, 2010) and analytical resonance and stiffness ratios for a tipless, picket-shaped cantilever. =0.05. Measurements fn, kHz
1
72.62
2
454.6
Kn, N/m
Qn
1.64
139
55.0
420
Jo
n
fn/f1
6.26
Kn/K1
33.54
Present analysis λn
n
1.923
3.11
4.81
104.1
fn/f1
Kn/K1
6.26
33.47
Table 8. Comparison of experimental (Exp) and analytical flexural resonance frequencies for a tipped picket-shaped cantilever. L=474 m, h=1.66 m, average width b=51.5 m, =2329 kg/m3, E=151 GPa. =0.01, =0.0606.
fn, kHz
fn/f1
n=1
2
3
4
5
6
Exp (Mendels, 2006)
9.978
62.21
174.8
343
567.5
848
FE (Mendels, 2006)
9.978
62.265
174.567
342.354
566.909
848.053
Present
9.987
62.229
173.339
337.849
555.447
825.147
Exp (Mendels, 2006)
6.235
17.519
34.376
56.875
84.987
Present
6.231
17.356
33.829
55.617
82.622
23
Table 9. Experimental results of tipped, picket-shaped Olympus AC240 cantilevers. Predictions of Eq. 5 and the Sader Method are presented in the last two columns. f1, kHz
Q1
K1, N/m
K1, Eq. 5.
K1, Sader Method
Li (2015)
77.55
242.098
2.18
2.141±0.108
3.42±0.099
66.286
129.972
1.79
1.564±0.079
1.5±0.043
Damircheli (2019)
76.114
142.6
2.93
2.062±0.104
1.97±0.057
Chawla (2011)
81
168
2.4
2.336±0.117
2.51±0.073
Shi (2017a)
80.51
184.1
2.22
2.308±0.116
2.73±0.079
Lai (2016)
65.6
90
1.6
1.532±0.077
1.02±0.029
Shi (2017b)
80.51
180
2.33
2.308±0.116
2.67±0.077
Saraswat (2013)
75.8
168
1.72
2.044±0.103
2.31±0.067
Sader (2014)
65.7
133
1.36
1.537±0.077
1.52±0.044
59.2
123
1.26
1.248±0.063
1.22±0.035
65.2
149
1.58
1.513±0.076
64.5
147
1.58
1.481±0.074
1.37
1.453±0.073
72.715
1.7
1.882±0.095
Thota (2007)
61.948
1.3
1.366±0.069
Solares (2012)
73.5
1.9
1.923±0.097
ro
63.88
Lai (2015)
1.68±0.049 1.64±0.047
Jo
ur na
lP
re
-p
Labuda (2016)
24
of
Reference
PII:
S0924-4247(19)32265-4
DOI:
https://doi.org/10.1016/j.sna.2020.111872
Reference:
SNA 111872
To appear in:
Sensors and Actuators: A. Physical
Received Date:
15 December 2019
Revised Date:
20 January 2020
Accepted Date:
24 January 2020
Please cite this article as: Mahmoud MA, Eigenvalues and dynamic stiffness of picket-shaped cantilevers, Sensors and Actuators: A. Physical (2020), doi: https://doi.org/10.1016/j.sna.2020.111872
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier.
Eigenvalues and dynamic stiffness of picket-shaped cantilevers M. A. Mahmoud Professor of Mechanical Engineering (Retired) 1222 Willow Point Dr, Murphy, Tx 75094, USA email: [email protected]
Highlights
Closed form expressions of the resonance frequency and dynamic stiffness are presented
ur na
lP
re
-p
ro
of
Graphical abstract
for picket-tapered cantilevers.
A new method for spring constant calibration is presented and verified.
Methods of application of the solution in sensing applications and design optimization are
Jo
outlined.
Abstract A comprehensive data set is presented that covers eigenvalues and dynamic stiffness of picketshaped microcantilevers for a wide range of the taper ratio and tip mass ratios. The shape of these microcantilevers is such that most of the length is of uniform cross section and the end segment of 1
the beam is width-tapered to a point at the free end. This configuration is widely used in MEMS devices. Closed form expressions are presented for the eigenvalues, and for static and dynamic stiffness. These expressions could help designers and researchers in the MEMS field in general, and atomic force microscopy in particular.
Keywords: Spring constant; Dynamic stiffness; Tapered cantilever; MEMS; Atomic force
of
microscopy; Calibration; Vibration; Resonance.
ro
1. Introduction
In many microelectromechanical (MEMS) devices, accurate knowledge of the resonance
-p
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,
re
mechanical properties, imaging self-assembled monolayers and determining the imaging or tapping force (Gibson, 2005), to name a few. The dynamic spring constants, as well as the entire
lP
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
ur na
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
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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.
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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. 2. Eigenvalues and dynamic stiffness of picket-shaped cantilevers For the picket-shaped microcantilever shown in Fig. 1, the resonance frequency “fn” of the
fn = λn2/(2π) √EI0 /(Mb L3 ),
= Lt /L,
and
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transverse vibration in the y direction, the taper ratio “” and the tip mass ratio “” are =Mtip /(Mb)
(1)
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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
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is the length of the tapered segment, Mtip is the tip mass, is the density, and Mb = b0 h L is the nominal cantilever mass. When = 0, the cantilever has a uniform cross section, and when =1
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the cantilever is fully tapered forming a triangular beam.
These two limiting cases were
investigated in (Mahmoud, 2020) in addition to trapezoidal-shaped beams.
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For partially tapered cantilevers, i.e. picket-shaped, there are no analytical solutions for the eigenvalues or the dynamic stiffness. Therefore, a numerical analysis based on the work of Myklestad (Mahmoud, 2019) was used in this study. In this analysis, the beam is discretized into
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a number of segments, the mass of each segment is lumped in the middle, and the beam is treated as a series of discrete masses connected by stiff, massless rods. This technique is also known in the literature as the Lumped Mass Transfer Matrix method. Details of how to best discretize nonuniform beams to ensure accurate results are given in (Mahmoud, 2019) where the accuracy of the
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Myklestad method’s predictions of the eigenvalues was validated for tapered beams with tip masses. The method was used in (Mahmoud, 2020) to determine closed form expressions of uniform and fully tapered microcantilevers with tip masses. For mode n, the dynamic stiffness ‘Kn’ (Melcher, 2007) and the normalized dynamic stiffness ‘n’ are 1
Kn = ∫0 EI() L [d2wn ()/d2 ]2 d / wn2 (1)
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and
n = Kn (L3/EI0)
(2)
wn() is beam deflection and EI() is the beam flexural rigidity, = x/L is the spatial coordinate along the beam (0 1). Eq. 2 is the result of equating expressions of the potential energy of the beam as a distributed mass system with the potential energy of the equivalent point mass system. Another dynamic property of interest is the equivalent mass, sometimes referred to as the effective mass. For mode n, the equivalent mass is the sum of the fraction (n) of beam mass that is equivalent to beam inertia effect plus the tip mass. It is denoted meq,n and is related to Kn by meq,n = (n + ) Mb;
n = λn4 (n + )
(3)
of
Kn = 4 π2 fn2 meq,n ;
The dynamic stiffness Kn should not be confused with the static stiffness “Ks”. The static stiffness,
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Ks, is the vertical static load that acts at the cantilever free end and causes a unit tip deflection. It is also widely known as the spring constant, and is not affected by the presence of a mass attached
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at the free end. For the uniform cantilever, Ks = 3 EI0/L3. On the other hand, the dynamic stiffness ‘Kn’ is mode related and represents the gradient of the tip dynamic force with respect to the tip
is 6
EI0
3 +2 L3
and
s =
6 3 +2
(4)
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Ks =
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deflection for a particular resonance mode “n”. For a picket-shaped cantilever, the static stiffness
s is the normalized static stiffness. The derivation is presented in the appendix.
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The author has refrained from using the term “effective mass” to describe the quantity meq,n. It was found (Mahmoud, 2016) that many researchers equate the “effective mass” to 0.24Mb (or sometimes 0.236 Mb or ¼ Mb). meq,n identically equals ¼ Mb only for uniform tipless cantilevers. The equivalent mass meq,n is dependent on the geometry, tip mass and vibration mode.
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Tables 1 and 2 present results of ‘λn’ and ‘n’ for = 0 - 0.06.
Table 3 shows the results of this study together with numerical results of Melcher, 2007 and Hahner, 2010 for both a uniform cantilever and a tipless picket cantilever of taper ratio = 25%. For the uniform cantilever, exact results obtained using the closed form solution in (Mahmoud, 2020) are included. As evident in the table, the present results coincide with the exact results for
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the uniform cantilever, and the present results are in good agreement with the published results for the picket cantilever.
Further evidence of the validity and accuracy of the solution is presented in the next two sections. It is very useful for both the designer and the analyst to have closed form expressions of the eigen
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values and dynamic stiffness for each mode over the practical ranges of and . For this purpose, an expanded database was generated using the transfer matrix numerical analysis in the ranges 0
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0.15 and 0 0.5. The results in the database were curve fitted and the resulting expressions
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are presented in Tables 4 and 5.
Fig. 2 shows examples of the variation trends of the normalized resonance frequencies (λ12 and
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λ22) and the dynamic stiffnesses (1 and 2) due to variations of the taper ratio and tip mass. It can be seen that as the taper ratio increases, the resonance frequencies increase, but the frequency
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ratios decrease. Presence of a tip mass decreases the frequencies and their ratios. Two opposing trends are observed for the dynamic stiffness. For the 1st mode: as increases, 1 increases to a maximum then decreases. The presence of a tip mass decreases 1. For the higher modes: n and
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n/1 decrease as increases and the presence of a tip mass leads to increases of both n and n/1.
3. Comparisons with experimental data
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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. 3.1 Frequency shift due to an attached mass at an arbitrary location Fig. 3 presents the shift of the fundamental resonance frequency, f1, due to the presence of a micro sphere (mass=6.5x10-12 kg) at a variable distance Lm from the fixed end of a picket 5
microcantilever with a taper ratio = 12% (λ1=1.992 and s =2.997). Given f1=38.75 kHz and Ks =20.15 N/m (Xie, 2008), and combining Eqs. 1 and 4, then Mb = (λ14 Ks)/(4π2 s f12) =1.784x10-9 kg (i.e. =3.64x10-3). Other variables are given in the caption of Fig. 3. Using the transfer matrix analysis, the eigen value with the mass attached “λ1,” was calculated for a number of micro sphere locations, and the frequency shift was calculated from f1= f1 {(λ1, / λ1)2 -1}. The results are shown as a solid line in Fig. 3 and are seen to be in good agreement with the experimental measurements
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3.2 Resonance frequency and stiffness ratios of tipless cantilevers
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reported by Xie, 2008.
A tipless picket cantilever was used for mass sensing (Lakshmoji, 2012) and its first six resonance frequencies are shown in the first row of Table 6. The taper ratio was unknown. Using the present
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analysis, the taper ratio =0.075 gives the best agreement with the frequency ratios fn/f1 as shown in the table. The largest difference between the present analysis and the experiment is 0.37%,
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indicating good agreement.
Another tipless microcantilever was analyzed next. It is a commercial probe, model FORTA-TL by Applied Nanostructures. From a photograph of the microcantilever, the taper ratio, , was
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determined to be 0.05. For the first 2 modes, the measured dynamic properties fn , Knd and Qn (Lozano, 2010) are shown in Table 7. Using = 0.05, calculated λn and n are shown in the table together with the calculated ratios f2/f1 and K2/K1. There is excellent agreement between the
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calculated and measured ratios.
3.3 Frequency and stiffness of picket cantilever with probing tips A picket-shaped silicon cantilever with a trapezoidal cross section was studied in detail in (Mendels, 2006), both experimentally and using finite elements (FE). The taper ratio 0.06, the tip mass ratio was unknown, and the dimensions and material properties are given in the caption of Table 8. The combination =0.0606 and =0.01 gives the best agreement with the experiment. Tables 4 and 5 were used for modes 1-4 and the transfer matrix analysis was used for modes 5 and 6
6. The table shows good agreement between the present analysis and both the experimental and FE results, especially for the lower modes. In (Mendels, 2006), the FE analysis that was based on a fully-clamed end of the beam did not agree well with the experiment. The FE analysis agreed with the experiment only after a segment of the wafer was included in the model to account for support flexibility. The present analysis is based on a clamped end, and hence the slight deviation of the present results from the experimental and FE results for the higher modes. From Eq. 4, Ks= 0.0834 N/m, which is in excellent agreement with the measured value Ks=0.0836
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N/m in (Mendels, 2006).
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4. Calibration of picket cantilevers
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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 As specified by the vendor (Olympus brochure
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microcantilever was used in this analysis.
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 m, respectively.
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The taper length Lt= 44 m; =18.3%. The tip mass ratio was selected as 0.02. =2330 kg/m3. E=190 GPa. It should be emphasized that the objective here is observing the qualitative trends of
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the results and, therefore, the particular cantilever dimensions and properties used in the numerical simulation do not significantly alter these trends. A dataset was generated by randomly changing all variables within 2% of their nominal values, and subsequently using the expressions of λ1 and 1 from tables 4 and 5 to calculate f1 and K1 for each combination of the variables.
The
equivalent mass parameter 1 was calculated by rearranging Eq. 3 in the form
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2 K1 = 1 f1,kHz ;
1= 4 π2 meq,1 x106
(5)
f1,kHz is the 1st mode resonance in kHz and meq,1 is the equivalent mass of mode 1. Fig. 4 presents the simulation results that show a linear trend of K1 vs f12 whereas the equivalent mass parameter 1 does not vary in any specific way with respect to either K1 or f1. These results suggest that small variations of dimensions and material properties lead to small, random variations in the equivalent mass. This observation offers a means of cantilever calibration: when 1 is determined 7
experimentally for a set of similar cantilevers, K1 may be calculated from Eq. 5 for other similar cantilevers.
To test the hypothesis that the equivalent mass parameter 1 could be considered almost constant (i.e. changing within a small scatter band) for nominally similar microcantilevers, experimental results for 26 individual Olympus AC240 cantilevers (tested in air) reported in 13 separate publications were compiled and used in the analysis. Table 9 presents experimental results of 16
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cantilevers. K1 is plotted in Fig. 5a (solid circles) against f12 for each of these cantilevers. Results of 10 more cantilevers are plotted in the figure as triangles. Each triangular symbol represents the
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averges of 13 results of each microcantilever (obtained by different researchers) as reported by Sader et al., 2016. The experimental variation of K1 vs f12 in Fig. 5a suggests a linear trend whereas
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1 does not vary in any specific way with respect to f1 as shown in Fig. 5b.
From the slope of the best line fit, Fig. 5a, it is concluded that for Olympus AC240 microcantilevers
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tested in air, 1 = 3.56 x10-4 1.79 x10-5. Predictions of K1 were obtained using this value of 1 in Eq. 5 and the results are presented in Table 9. For completeness, the Sader Method (Sader et
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al. 2016) was also used to obtain estimates of K1 as shown in the last column of the table. Table 9 shows that the results of Eq. 5 are in reasonable agreement with the experiments and in some cases are in better agreement with experiments than the Sader method predictions. One
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principal advantage of Eq. 5 is that it only requires measuring f1. Sader’s method requires measuring both f1 and the quality factor Q1. However, the Sader method is more general because including Q1 accounts for effects of ambient fluid properties such as density and viscosity. Nevertheless, the results of Eq. 5 serve as a strong evidence of the validity of the expressions of λ1
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and 1 presented in Tables 4 and 5.
A significant advantage of Eq. 5 is that the calibration of the cantilever does not require tip-surface contact which is required for many methods of spring constant calibration. Tip-surface contact can be a disadvantage as it may cause changes in the tip shape of the AFM probe which will have significant effects on image resolution, measured adhesion and measured sample mechanical properties. Studies by Slattery et al, 2013 and Song et al, 2015 discuss how tip-surface contact 8
can change tip shape as well as various methods of spring constant and detector sensitivity calibration that do not require tip-surface contact. The Sader Method utilizes the formula K1 = C1 Q1 f11.3 , where C1 is a calibration coefficient. Both Eq. 5 and the Sader Method are based on the implicit assumption that effects of factors such as support compliance, undercuts and thickness variation are included within the scatter ranges of 1
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and C1.
One aspect not addressed in this study is the effect of imaging tip set-back observed in many
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commercial AFM cantilevers. Fortunately, the Olympus AC240 microcantilevers studied herein have their imaging tips right at their free ends. For other microcantilevers, there is a need for further investigation of the effect of tip set-back. This is important in view of the cubic dependence
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of Kn on the distance “L” between the base of the cantilever and the imaging tip, as in Eq. 2. This effect is particularly important for short cantilevers where the tip set-back could be significant
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relative to the length (Slattery et al, 2014, 2019). Theoretical work would be useful, but also experimental research is needed. Methods of cantilever calibration that can determine the spring
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constant at specific points or positions along the length of cantilevers of different geometries are required. The inverted reference cantilever method described by Slattery et al, 2013 may be useful for such experiments. Also, the added mass method could be used by changing the position of the
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added mass along the length of cantilever as proposed by Vakarelski et al, 2007. Until further research into the tip set-back effect is conducted, an approximate adjustment factor that accounts for this effect proposed by Sader et al., 2012 may be utilized. It should be noted that in addition to calibration of the cantilever spring constant, there are a
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number of other important calibrations that must be done to the entire AFM system for quantitative measurements and these include the cantilever detector sensitivity and the piezoelectric actuator. A paper by Wagner et al, 2011 discusses these calibrations and provides a good overview. 5. discussion and conclusion To illustrate how the closed form expressions in Tables 4 and 5 could be useful in tuning cantilever design to achieve desired characteristics, reference is made to a recent study (Keyvani, 2017). In 9
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 the speed of the tip is reduced right before touching the sample surface. Hence, the proposed cantilever indents the sample less and, therefore, applies a lower force. Their proposed cantilever was linearly tapered along the whole length such that the width at the free end is smaller than the
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width at the clamped end.
It is possible to achieve the same goal, i.e. f2/f1=6, using a picket-shaped cantilever. Fig. 6 is a
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plot of some of the present results and shows the correlations of , , f2/f1 and 2/1. Intersections of a vertical line through f2/f1= 6 with the grid of and curves yield a variety of and
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combinations to obtain f2/f1 =6. Some of these (, ) combinations are (0.2, 0.01), (0.2, 0.126), (0.25, 0), (0.17, 0.022) and (0.222, 0.0045). It is evident from the figure that the ratio f2/f1 = 6 is
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not attainable for 0.15.
The present analysis could also be useful in sensing applications. As an example, consider two
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cantilevers of identical dimensions and material properties differing only in that one is uniform and the other is picket-shaped with =0.25. When a particle of (normalized) mass adsorbs to the apex of the uniform cantilever, the sensitivity of mass sensing is Su ∝ (λ2n,after − λ2n,before )u /.
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When an identical particle adsorbs to the apex of the tapered beam, the sensitivity of mass sensing is Stapered ∝ (λ2n,after − λ2n,before )tapered /. The ratio of the average mass sensitivities between the two cantilevers is Stapered/Su =
(λ2n,after −λ2n,before )tapered (λ2n,after −λ2n,before )u
. As an example, for mode 2 and =0.001 and
using table 4, Stapered/Su = (5.18522-5.22)/(4.68942-4.6942) 3.6. For the 1nd and 3rd modes, the
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tapered beam’s sensitivity is higher than the corresponding uniform beam’s sensitivity by a factor of 2.1 and 5.2, respectively. In conclusion, closed form expressions of the resonance frequency and dynamic stiffness, based on results of numerical analysis, were presented for picket-shaped cantilevers with tip masses. These cantilevers are widely used in MEMS devices as they offer the possibility of tailoring the resonance frequency and the dynamic stiffness to suit different applications. Picket-shaped cantilever beams are softer (more compliant) but have larger resonance frequencies than uniform 10
beams. The resonance frequencies and stiffness vary in different patterns depending on the taper ratio, the tip mass and mode order. The closed form expressions presented in Tables 4 and 5 agree well with experiments.
Author statement
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I declare that I am the sole author. I declare that this research was not funded by any organization.
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M.A. Mahmoud
Appendix: Static spring constant of picket-shaped cantilevers
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A plan view of a picket-shaped cantilever is shown in Fig. A-1. The bending moment ‘Mz(x)’ at
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distance x due to a force ‘F’ normal to the plane of drawing acting at x=L is Mz(x) = F (L-x). Beam width, bx, in the tapered segment at distance x (see Fig. A-1) is bx = b0 (L-x) / L; EI(x) = EI0 (L-x) / L.
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In the range 0x L (the vertical deflection is denoted y1): y1`` (x) = F(L-x)/ EI0 ; y1` (x)= F (L x - x2/2) / EI0 ; y1(x)= F (L x2/2 – x3/6) / EI0
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In the range L x L (notice that + = 1 and the vertical deflection is denoted y2), y2`` (x) = F(L-x)/ EI(x) = F L / EI0 ; y2` (x) = F Lx / EI0 + C ; y1` ( L) = y2` ( L) yields C = ½F 2L2 / EI0 y1( L) = y2( L) yields D = – F 3L3 / 6EI0 y2(L)= F ( L 3/2) / EI0 + C L + D = {(3+2)/6}{FL 3/ EI0}
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y2(x)= ½F L x2 / EI0 + C x + D
Ks =
F y2 (L)
=
6
EI0
3 +2
L3
, and s =
6 3 +2
is the normalized static stiffness.
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Labuda, A., Kocun, M., Lysy, M., Walsh, T., Meinhold, J., Proksch, T., Meinhold, W., Anderson, C., Proksch, R., 2016. Calibration of higher eigenmodes of cantilevers. Rev. Sci. Instrum. 87, 073705. Lai, C-Y, Barcons, V., Santos, S., Chiesa, M., 2015. Periodicity in bimodal atomic force microscopy. J. Appl. Phys. 118, 044905.
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Li, C., 2015. Bimodal AFM Using State Estimation and In Vitro AFM Imaging of Microtubules. Master of Science Thesis, Department of Electrical Engineering, University of Minnesota. Lozano, J., Kiracofe, D., Melcher, J., Garcia, R., Raman, A., 2010. Calibration of higher eigenmode spring constants of atomic force microscope cantilevers. Nanotechnology 21, 465502.
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Mahmoud, M.A., 2016. Validity and accuracy of resonance shift prediction formulas for microcantilevers: a review and comparative study. Crit. Rev. Solid State Mater. Sci. 41, 386. Mahmoud, M.A., 2019. Natural frequency of axially functionally graded, tapered cantilever beams with tip masses. Eng. Struct. 187, 34. Mahmoud, M.A., 2020. Dynamic spring constants of higher resonance modes of uniform and tapered microcantilevers with tip masses. Micron 129, 102795. Melcher, J., Hu, S., Raman, A., 2007. Equivalent point-mass models of continuous atomic force microscope probes. Appl. Phys. Lett. 91, 053101.
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Mendels, D.A., Lowe, M., Cuenat, A., Cain, M.G., Vallejo, E., Ellis, D., Mendels, F., 2006. Dynamic properties of AFM cantilevers and the calibration of their spring constants. J. Micromech. Microeng. 16, 1720. Sader, J.E. et al, 2016. A virtual instrument to standardise the calibration of atomic force microscope cantilevers, Rev. Sci. Instrum. 87, 093711. See also https://sadermethod.org/ Sader, J.E., Friend, J.R., 2014. Note: Calibration of atomic force microscope cantilevers using only their resonant frequency and quality factor. Rev. Sci. Instrum. 85, 116101. Sader, J.E., Sanelli, J.A., Adamson, B.D., Monty, J.P., Wei, X., Crawford, S.A., Friend, J.R., Marusic, I., Mulvaney, P., Bieske, E.J., 2012. Spring constant calibration of atomic force microscope cantilevers of arbitrary shape. Rev. Sci. Instrum. 83, 103705. Saraswat, G., Salapaka, M.V., 2013. On contribution and detection of higher eigenmodes during dynamic atomic force microscopy. Appl. Phys. Lett. 102, 173108.
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Shi, S., Guo, D., Luo, J., 2017a. Interfacial interaction and enhanced image contrasts in higher mode and bimodal mode atomic force microscopy. RSC Adv. 7, 55121. Shi, S., Guo, D., Luo, J., 2017b. Imaging contrast and tip-sample interaction of noncontact amplitude modulation atomic force microscopy with Q-control. J. Phys. D: Appl. Phys. 50, 415307. Slattery, A.D., Blanch, A.J., Quinton, J.S., Gibson, C.T., 2013. Calibration of atomic force microscope cantilevers using standard and inverted static methods assisted by FIB-milled spatial markers. Nanotechnology 24, 015710. Slattery, A.D., Blanch, A.J., Ejov, V., Quinton, J.S., Gibson,C.T., 2014. Spring constant calibration techniques for next-generation fast-scanning atomic force microscope cantilevers. Nanotechnology 25, 335705. Slattery, A.D., Blanch, A.J., Shearer, C.J., Stapleton, A.J., Goreham, R.V., Harmer, S.L., Quinton, J.S., Gibson, C.T., 2019. Characterisation of the Material and Mechanical Properties of Atomic Force Microscope Cantilevers with a Plan-View Trapezoidal Geometry. Applied Sciences 9, 2604.
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Xie, H., Vitard, J., Haliyo, S., Régnier, S., 2008. High-sensitivity mass and position detection of micro-objects adhered to microcantilevers. J. of Micro-Nano Mechatronics 4, 17.
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Author Biography 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.
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He retired in 2018.
Fig. 1. Picket-shaped microcantilever.
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y L Lt
Mtip
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= Lt / L
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x
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z
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Fig. 2. Variation trends of the normalized resonance frequencies and dynamic stiffness due to variations of taper ratio and tip mass.
λ2 2
4 0.1 0.2 0.3 0.4 0.5
3.2
1
3
s
25 20 180 0
3 0
f2/f1
5
6.5
2
130 80 30
2.8 0
Taper ratio
0.1 0.2 0.3 0.4 0.5
35 10
0
0.1 0.2 0.3 0.4 0.5
5.5 4.5 60 0
0.1 0.2 0.3 0.4 0.5
2/1
λ12
30
0.1 0.2 0.3 0.4 0.5
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6
0
Taper ratio
0.1 0.2 0.3 0.4 0.5
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Taper ratio
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Fig. 3. Frequency shift, f1, due to the presence of a micro sphere at Lm. The symbols are experimental results of Xie, 2008. Taper ratio =0.12. f1=38.75 kHz, L=598 m, b0=141m, added mass=6.5x10-12 kg. Static spring stiffness Ks=20.15 N/m and =3.64x10-3.
Analysis Series1
(Xie, 2008) Series2
0
-200
L
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f1 , Hz
-100
Lm -400 0.2
0.4
0.6
0.8
1
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0
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-300
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Lm /L
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Fig. 4. Simulation results. Variation of K1 vs f12 due to 2% random errors in all variables results in a linear trend of K1 vs f12 whereas 1 does not vary in any specific way with respect to K1 and f1. =0.02, =18.3%. 0.0004
1.6
1
1.4
0.0003 1.1
1.3
1.2
1.3
1.4
1.5
1.6
K1, N/m
1.2
0.0004
1.1
1
1 0.9 3000
0.0003 3500
4000 2
56
4500
58
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K1, N/m
1.5
60
62
f1, kHz
2
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f1 , kHz
17
64
66
Fig. 5. (a) Experimental variation of K1 vs f12 for 26 Olympus AC240 microcantilevers. (b) The equivalent mass parameter 1 = K1/f12 does not vary in any specific way with respect to f1. 3.5
(a)
3
(b) 0.0006
2
1
1.5
0.0004 0.0002
1
2 K1 =0.000356 f1,kHz
0.5 0 3000
0 55
65
75
of
K1, N/m
2.5
f1, kHz
4000
5000
6000
7000
Jo
ur na
lP
re
-p
ro
f12, kHz2
18
85
Fig. 6. The resonance and stiffness ratios (modes 2 and 1) as functions of the taper ratio (continuous lines) and tip mass ratio (dotted and dashed lines). K2/K1 = 2/1.
80 70 = 0.15
50
0.12
40
0.10 0.08
30
of
K2 / K1
60
20
ro
10 0 5.25
5.5
5.75
6
6.25
6.5
Jo
ur na
lP
re
-p
f2 / f1
19
Fig. A-1. Plan view of a picket-shaped cantilever.
L x +=1
bx
b0
x L
Jo
ur na
lP
re
-p
ro
of
L
20
Table 1. Eigen values ‘λn’ and normalized dynamic stiffness ‘n’ for =0 and =0.02. =0
λ1
= 0.02 λ2
λ3
λ4
1
2
3
4
λ1
λ2
λ3
1
λ4
2
3
4
1.875 4.694 7.855 10.996
3.091 121.4 951.6 3654
1.839 4.610 7.722
10.820 3.077 138.4 1223 5298
0.051
1.923 4.810 8.044 11.253
3.111 104.0 714.1 2417
1.883 4.701 7.848
10.962 3.092 118.6 923.9 3579
0.101
1.972 4.922 8.212 11.462
3.131 90.64 560.2 1724
1.927 4.782 7.940
11.026 3.108 103.1 723.5 2591
0.149
2.020 5.021 8.347 11.607
3.151 80.39 453.7 1284
1.970 4.849 7.988
11.006 3.121 90.93 583.7 2017
0.2
2.072 5.117 8.459 11.700
3.172 71.21 365.7 957.0
2.015 4.906 7.995
10.920 3.133 79.92 474.0 1705
0.222
2.095 5.155 8.497 11.722
3.180 67.62 333.6 851.2
2.035 4.926 7.984
10.873 3.137 75.60 436.6 1638
0.25
2.123 5.200 8.535 11.737
3.189 63.42 298.2 744.9
2.060 4.947 7.962
10.819 3.140 70.61 399.8 1614
0.278
2.152 5.241 8.563 11.743
3.198 59.50 267.5 662.7
2.085 4.963 7.930
10.769 3.142 65.97 370.0 1617
0.303
2.178 5.274 8.581 11.744
3.205 56.15 243.6 604.4
2.107 4.973 7.898
10.733 3.142 62.11 349.8 1646
0.333
2.210 5.310 8.594 11.744
3.211 52.40 219.4 549.4
2.134 4.980 7.858
0.379
2.257 5.355 8.602 11.746
3.215 47.30 190.8 486.7
2.173 4.980 7.798
0.422
2.302 5.387 8.603 11.752
3.213 43.05 170.4 439.7
2.209 4.972 7.749
0.5
2.381 5.425 8.602 11.760
3.191 36.68 144.1 371.2
2.272 4.940 7.680
ro
of
0
10.699 3.140 57.90 332.8 1702 10.657 3.130 52.37 316.0 1780
-p
10.628 3.114 48.04 309.3 1863 10.573 3.063 42.06 306.6 2000
= 0.4
= 0.06
λ2
λ3
λ4
1
2
3
4
λ1
λ2
λ3
λ4
1
2
3
4
1.807 4.542 7.625 10.704
3.066 157.3 1551 7428
1.776 4.486 7.552 10.624 3.058 178.1 1932 10026
0.051
1.846 4.614 7.713 10.787
3.078 135.4 1199 5304
1.813 4.545 7.615 10.674 3.067 154.3 1531 7539
0.101
1.886 4.675 7.761 10.790
3.090 118.3 965.5 4156
1.850 4.591 7.639 10.651 3.076 135.8 1272 6315
0.149
1.925 4.721 7.767 10.724
3.098 104.9 808.9 3613
1.884 4.622 7.624 10.575 3.081 121.4 1106 5922
0.2
1.965 4.754 7.732 10.621
3.105 92.85 695.9 3513
1.920 4.640 7.574 10.481 3.084 108.6 999
6218
0.222
1.983 4.763 7.706 10.578
3.106 88.17 660.6 3580
1.936 4.643 7.545 10.445 3.083 103.7 970
6518
0.25
2.005 4.770 7.670 10.536
3.106 82.87 630.8 3770
1.955 4.642 7.507 10.413 3.081 98.25 953
7069
0.278
2.026 4.772 7.629 10.500
3.104 78.00 609.8 3999
1.974 4.637 7.467 10.388 3.077 93.28 947
7677
0.303
2.045 4.770 7.592 10.478
3.100 74.03 600.3 4252
1.991 4.629 7.434 10.374 3.071 89.31 955
8300
0.333
2.068 4.764 7.552 10.461
3.094 69.82 599.0 4586
2.011 4.617 7.399 10.366 3.062 85.21 979
9092
0.379
2.101 4.746 7.498 10.439
3.077 64.43 608.1 5044
2.039 4.591 7.355 10.354 3.041 80.10 1031 10177
0.422
2.131 4.723 7.460 10.424
3.053 60.45 629.9 5474
2.065 4.564 7.327 10.347 3.014 76.60 1099 11177
0.5
2.182 4.674 7.411 10.392
2.990 55.41 680.6 6183
2.105 4.513 7.293 10.325 2.945 72.79 1236 12839
ur na
0
Jo
λ1
lP
re
Table 2. Eigen values ‘λn’ and normalized dynamic stiffness ‘n’ for =0.04 and =0.06.
Table 3. Normalized dynamic stiffness ‘n’ and equivalent mass ‘βn’ for =0. =0 n
βn
=0.25 n
21
βn
βn,a *
n
Present Exact Melcher† Hahner‡ Present Exact Melcher Hahner Present Melcher Hahner Present Present Melcher Hahner
1
3.096
2 3
3.096 3.09
3.09
0.25
0.25 0.249
0.25
3.19
3.21
3.21
0.157
0.179 0.178
0.18
121.38 121.38 120.6
120.6
0.25
0.25 0.248
0.248
63.41
63
63.9
0.087
0.099 0.098
0.1
951.64 951.64 936
939
0.25
0.25 0.245
0.247
298.2
290.1
302.4
0.056
0.064 0.062
0.065
* βn,a is the effective mass normalized relative to the actual mass of the picket cantilever; βn = βn,a (1-/2). † Melcher et al., 2007. ‡ Hahner, 2010.
Table 4. Expressions for the eigenvalues ‘λn’ in the range 0 0.15 and 0 0.5. =10 . ψ =10 .
0.23%
0.12 %
Expression
of
0.16%
†
λ1 = (1.8751-1.85+4.12- 6.4 3) (0.0732 +0.5092 +1) {1- (2.4 2 + 1.8 ) +(18 3 + 8.5 2 + 5.9 ) 2 + (-55.5 3-17.32- 11) 3} λ2=4.694 -0.465 +0.247 2-0.094 3+0.017 4 +( 0.231-0.29 +0.3042-0.2433+0.114 4-0.0223 5) ψ +( -0.0013 -0.0183 0.062+ 0.09663-0.0414 4-0.00367 5+.00444 6) ψ2 +( -0.005 -0.0181 + 0.05272 -0.03593-0.0103 4+0.02057 50.00613 6) ψ3 +( 0.00038+ 0.002198 -0.002628 2 -0.003673+0.00895 4-0.0061246 5+0.00142 6) ψ4
ro
Accuracy*
(0.15 > > 0.085)∩(0.5 > > 0.22)
0.37%
-p
λ3= 9.88 2 - 5.11 + 8.372 +(1134 4 - 652.16 3 + 143.62 2- 14.2 + 0.135) ψ+(-3.19 2 + 0.963 - 0.001) ψ2+(-2.88 3 + 1.3033 2- 0.20763 + 0.0072) ψ3 Full range except (0.15 > >0.085)∩(0.5> >0.22)
0.21%
re
λ3=7.845-0.819+ 0.955 2-0.9 3 + 0.466 4-0.095 5+( 0.46-0.343 -1.078 2+ 2.162 3-1.39 4+ 0.303 5) ψ+( -0.0835-0.5354 + 1.761 2-2.189 3+ 1.1937 4 -0.239 5) ψ2 +( 0.00414+ 0.0694 -0.1717 2+ 0.1716 3-0.0714 4 + 0.00954 5) ψ3 <0.033
lP
λ4= χ {1+(-1 -1.624 ψ+1.116 ψ2-1.382 ψ3+0.4857 ψ4-0.0706 ψ5+0.003688 ψ6) +(11.6+29.1 ψ -7.12 ψ2-19.11 ψ3+23.95 ψ4-6.391 ψ5+0.5225 ψ6) 2+(-103.8+336.2ψ-2525 ψ2 +3830 ψ3-2094.3 ψ4+435.08 ψ5-31.33 ψ6) 3+(553.5-13215ψ+54487 ψ2-70977 ψ3+34586 ψ4-6750 ψ5+467 ψ6) 4} χ = -172.02 5 + 219.29 4 - 82.9 3 + 0.8149 2 + 5.1869 + 10.996 0.22%
0.15>>0.026
ur na
λ4= χ {1- 0.114 + 2.64 2-31.47 3+122.8 4-203.5 5+ 123.7 6+( -0.85 -5.29 -122.12+ 882.6 3 -1990.5 4+ 1493.3 5) + (7.09+ 24.4 + 2072.52-12884 3+27348 4-19761 5) 2+( -33.7-2.17 -13933 2+81260 3-167187 4+118236 5) 3+(66.9193.4 + 336152-189217 3+381833 4-266209 5) 4}
* Largest percent difference between fitted and numerical results. † The accuracy is 0.16% in the range {0 0.1 and 0 0.42}, 0.22% for {0.1< 0.15 and 0 0.42}, and 0.4% for 0.5> > 0.42.
Table 5. Expressions for the normalized dynamic stiffness ‘n’ in the range 0 0.15 and 0 0.5. =10 . ψ =10 . = 100 . Accuracy
1 = (3.0904-0.73+3.62-83)+ (0.38-5.1+40.62-1923+3954) +(0.33-3.4+9.62) 2+ (-0.76-52+6602-36403+76304) 3 +(-1.27+39+192-5,6153+47,8904-125,0005) 4
Jo
0.07%
Expression
0.18%
<0.033 2 = (121.375+79.65 +28 2-6.92 3+3.7 4)+(-395-281 +162 2-159 3-56 4+213 5) +(1142+476 -419 2+707 3+247 4) 2 +(-2,818-3,208 +8,275 2-1,084 3-38,098 4+52,044 5) 3 +(4,010+8,663 -12,742 2-53,763 3+27,9831 4-499,532 5+339,519 6) 4 +(-2,276-6,931 +4,544 2+62,910 3-240,080 4+363,896 5-209,541 6) 5
0.32%
>0.032 2 =(121.286+80.5 +25 2-2.52 3+0.48 4)+(-397-259.4 +83.6 2-66 3+29.5 4-5.48 5) +(1173+190 +478.7 2-181.9 3+32.6 4) 2+(-3,067-922 +1,992 2-3,744 3+2,283 4-496 5) 3 +(4,200+7,874 -22,108 2+34,629 3-27,549 4+11,271 5-1,874 6) 4 +(-2,315-7,844 +20,542 2-29,813 3+22,877 4-9,150 5+1,498 6) 5
0.32%
<0.036
22
3 = (951.61+1208 +779 2-201 3+248 4-282 5+188.5 6)+(-594.17-776 +557 2-515.7 3-3599 4+12856 5-13015 6) ψ +(300.37+267.6 -420 2-1081 3+16260 4-46055 5+43624 6) ψ2 +(-123.053-203 +750 2+1297 3-18714 4+50829 546817 6) ψ3 +(31.84+91.2 -238.43 2-1305.6 3+10278 4-25005 5+21870 6) ψ4 +(-4.3482-17.177 +26.96 2+349.3 32164.8 4+4904.3 5-4124 6) ψ5 +(0.2383+1.146 -0.8457 2-28.71 3+156.27 4-336.8 5+274.6 6) ψ6 >0.035 3 = (948.478+1229.4 +700.8 2-39.3 3+8.06 4)+(-567.5-670.76 +74.03 2+1.486 3-16 4+5 5) ψ +(220.294+131 +213.82 2-123.83 3+32 4) ψ2 +(-72.08+84.52 -285 2+287.63 3-143.3 4+28.031 5) ψ3 +(13.93-32.856 +97.88 2-120.237 3 +86.9 4-33.61 5+5.39 6) ψ4 +(-0.993+2.532 -7.263 2+8.386 3-5.922 4+2.251 5-0.356 6) ψ5
0.59%
(<0.035)∩(>0.149) 4 =(2819.209-547.3 +1724.234 2-1373.32 3+486.53 4-86.611 5+6.218 6)+(-1056.1+2512.4 -3809 2+2986.95 3-1011.57 4+172.439 5-11.842 6) ψ +(-266.04-3008.1 +3529.61 2-2554.266 3+814.55 4-130.774 5+8.399 6) ψ2 +(290.016+1582.76 -1575.62 2+1074.497 3-321.332 4+47.8518 5-2.782445 6) ψ3 +(-80.893-422.11 +377.567 2244.237 3+68.729 4-9.4308 5+0.480932 6) ψ4 +(9.9374+56.172 -46.609 2+28.746 3-7.6299 4+0.9557 5-0.04042 6) ψ5 +(-0.465-2.9693 +2.3288 2-1.37537 3+0.34474 4-0.03888 5+0.001214 6) ψ6
0.59%
Full range except (<0.035)∩( >0.149) 4 = (3,653.433+6,945 +6,550 2-1,135 3+921 4-411 5+75.7 6)+(-3,281-5,385 -856 2+3,144 3-3,287 4+1,630 5-314 6) ψ +(2,083-449 +11,445 2-18,423 3+16,507 4-7,534 5+1,376 6) ψ2 +(-970.6+1,592 -6,757 2+9,251 3-6,332.6 4+2,113.4 5-262 6) ψ3 +(279.92-543.9 +1,805.5 2-1,274.45 3-76.7 4+523.2 5-174 6) ψ4 +(-42.42+72.75 -221.77 266.47 3+324.7 4-254.86 5+63.73 6) ψ5 +(2.556-3.43 +9.977 2+18.2 3-35.95 4+24.33 5-5.723 6) ψ6
ro
of
0.32%
1
2
3
Experimental fn, kHz
12.5945
78.471
219.36
6.231
17.417
Experimental fn/f1 Analytical λn
1.9467
4.8651
Analytical fn/f1 = λn / λ1
6.246
Difference
0.24%
2
5
6
429.272
708.084
1,056.398
34.084
56.222
83.878
8.1284
11.3606
14.5835
17.796
17.435
34.057
56.121
83.569
0.10%
-0.08%
-0.18%
-0.37%
lP
2
4
re
Mode n
-p
Table 6. Comparison of experimental (Lakshmoji, 2012) and analytical resonance ratios for a tipless, picket-shaped cantilever. =0.075.
ur na
Table 7. Comparison of experimental measurements in (Lozano, 2010) and analytical resonance and stiffness ratios for a tipless, picket-shaped cantilever. =0.05. Measurements fn, kHz
1
72.62
2
454.6
Kn, N/m
Qn
1.64
139
55.0
420
Jo
n
fn/f1
6.26
Kn/K1
33.54
Present analysis λn
n
1.923
3.11
4.81
104.1
fn/f1
Kn/K1
6.26
33.47
Table 8. Comparison of experimental (Exp) and analytical flexural resonance frequencies for a tipped picket-shaped cantilever. L=474 m, h=1.66 m, average width b=51.5 m, =2329 kg/m3, E=151 GPa. =0.01, =0.0606.
fn, kHz
fn/f1
n=1
2
3
4
5
6
Exp (Mendels, 2006)
9.978
62.21
174.8
343
567.5
848
FE (Mendels, 2006)
9.978
62.265
174.567
342.354
566.909
848.053
Present
9.987
62.229
173.339
337.849
555.447
825.147
Exp (Mendels, 2006)
6.235
17.519
34.376
56.875
84.987
Present
6.231
17.356
33.829
55.617
82.622
23
Table 9. Experimental results of tipped, picket-shaped Olympus AC240 cantilevers. Predictions of Eq. 5 and the Sader Method are presented in the last two columns. f1, kHz
Q1
K1, N/m
K1, Eq. 5.
K1, Sader Method
Li (2015)
77.55
242.098
2.18
2.141±0.108
3.42±0.099
66.286
129.972
1.79
1.564±0.079
1.5±0.043
Damircheli (2019)
76.114
142.6
2.93
2.062±0.104
1.97±0.057
Chawla (2011)
81
168
2.4
2.336±0.117
2.51±0.073
Shi (2017a)
80.51
184.1
2.22
2.308±0.116
2.73±0.079
Lai (2016)
65.6
90
1.6
1.532±0.077
1.02±0.029
Shi (2017b)
80.51
180
2.33
2.308±0.116
2.67±0.077
Saraswat (2013)
75.8
168
1.72
2.044±0.103
2.31±0.067
Sader (2014)
65.7
133
1.36
1.537±0.077
1.52±0.044
59.2
123
1.26
1.248±0.063
1.22±0.035
65.2
149
1.58
1.513±0.076
64.5
147
1.58
1.481±0.074
1.37
1.453±0.073
72.715
1.7
1.882±0.095
Thota (2007)
61.948
1.3
1.366±0.069
Solares (2012)
73.5
1.9
1.923±0.097
ro
63.88
Lai (2015)
1.68±0.049 1.64±0.047
Jo
ur na
lP
re
-p
Labuda (2016)
24
of
Reference