iMechanica - Microcantilever - Comments

Surface stress effect on shear modulus

hamed hamed — Fri, 04 Apr 2008 13:04:00 -0400

does surface stress enfluence on shear modulus?

resonance (natural) frequency of a cantilever beam

Somashekara Bhat — Sat, 03 Nov 2007 07:58:03 -0400

Thanks and regards
Somashekara Bhat

resonance (natural) frequency of a cantilever beam

Somashekara Bhat — Sat, 03 Nov 2007 07:56:39 -0400

Thanks and regards
Somashekara Bhat

Natural frequency....

ramdas chennamsetti — Fri, 02 Nov 2007 14:14:51 -0400

R. Chennamsetti, Scientist, R&D Engineers, India

The procedure is as following.

(a) Derive governing equation of a beam for lateral vibration. You need to use Newton's second law. This will be fourth order in space and second order in time.

(b) Assume there is no external loading on the beam. Use separation of variables technique. We get two separate equations, one is in space and the other in temporal domain. Both equal to some constant (this is sqaure of circular frequency).

(c) In spatial domain, the equation is fourth order. Four constants appear in the solution. Use four boundary conditions of beam, two at each end. For a cantilever (fixed-free) beam, in first mode (1.875)^2 = 3.52 (approx) appears.

(d) In time domain, the equation is second order. Two initial conditions are required.

(e) f = {(1.875)^2/2pi}sqrt(EI/(m*L^3)) = {(1.875)^2/2pi}sqrt(EI/(rho*A*L^4))

You may check your term in the denominator, which is in square root.

For more details refer the following.

S. S. Rao, 'Mechanical Vibrations' 4th edition, Pearson Edition, Page - 609 - 613.

How to get the flexural vibration frequency of a beam

Ji Wang — Fri, 02 Nov 2007 02:45:36 -0400

Indeed this is a simple procedure. You can start from the flexural vibration equation of a beam, then applying the boundary conditions (one fixed end and one free end). You can find the complete procedure in books by Timoshenko like the Vibration one.

Also many books on elementary vibration and structural analysis should have the procedure given.

very well received

Rui Huang — Sun, 29 Apr 2007 11:38:41 -0400

Kilho:

Thanks for your response. I appreciate it, especially the list of references that seem to be interesting and relevant.

Curiously I wonder what is the current understanding of the low quality factors in micron-scale oscillators and what one can do to improve the quality factor in liquid. I think the related mechanics is quite interesting, probably requiring studies on both solid and fluid mechanics.

RE: cantilever operated in liquid for biomolecular detection

Kilho Eom — Sun, 29 Apr 2007 02:40:47 -0400

Thank you for your interests to my current work. As you mentioned, microcantilever is operated in the flexural mode in liquid. The resonance behavior of a cantilever in liquid is well described in the reference by Kirstein (S. Kirstein, M. Mertesdorf, M. Schonhoff, J Appl Phys, 84, p1782, 1998), which I used in the paper. For the effect of compressional wave, I still need to check how it affects the resonance of a cantilever. The resonance of a cantilever in liquid might be also dependent on the liquid cell size, but in this paper we did not consider.

For quality factor, obvious Q=15 in liquid is not quite high. But, most of micron-scale cantilever exhibits the low quality factor in liquid environment, so that it is very difficult to detect the biomolecule in liquid based on the resonant frequency shift due to biomoelcular interactions (e.g. protein antigen-antibody interaction, DNA hybridization). For instance, Craighead group have recently reported the micron-scale oscillator which exhibit the high-frequency dynamical range (e.g. 20 ~ 100 MHz) as well as high Q factor (e.g. Q ~ 10^6) in normal air [for details, See paper by Verbridge et al., Nano Lett, 6, p2109, 2006]. However, their oscillator exhibit the very poor Q factor (e.g. Q ~ 5) in liquid environment, so that their oscillator may not be used as an in-situ biosensor yet. Until recently, as far as I know, most micron-scale cantilever resonator possesses the low Q factor in liquid so that it has not been well used as an in-situ biosensor. Hence, I believe that the challenge resides in the development of micron-scale (or nano-scale) oscillator that bears the high Q factor even in liquid environment for in-situ biosensor applications.

For PZT materials, we could not give the detailed information in the paper because of length. For our PZT resonating cantilever, we used PZW-PZT thick film, which is well described in the following references: T.Y. Kwon, et al., J Cryst Growth, 295, p172, 2006; T.Y. Kwon et al, Appl Phys A, in press. Also, the fabrication method of microcantilever is well described in the following paper: J.H. Park et al, Adv Funct Mater, 15, p2021.

I think that NEMS/MEMS resonators may be interesting to people in mechanics and/or applied physics, because of their various applications. The following references might be interesting.

S.B. Shim, M. Imboden, P. Mohanty, "Synchronized oscillation in coupled nanomechanical resonators," Science, 316, p95 (2007)

M. Li, H.X. Tang, M.L. Roukes, "Ultra-sensitive NEMS-based cantilevers for sensing, scanned probe, and very high-frequency applications," Nature Nanotechnol., 2, p114 (2007)

J.S. Bunch, A.M. van der Zande, S.S. Verbridge, I.W. Frank, D.M. Tanenbaum, J.M. Parpia, H.G. Craighead, P.L. McEuen, "Electromechanical resonators from graphene sheets," Science, 315, p490 (2007)

vibrations in liquid environment

Rui Huang — Thu, 26 Apr 2007 14:17:15 -0400

Interesting work. I am sure many imechanicians who have wroked on fluid-structure interactions would find this an interesting application in biomolecular research. I have not really worked much in this area myself, but would like to share some experience about vibrations in a liquid environment.

Since early 1980s, many liquid-phase sensors/detectors (mostly for chemical applications; see Ref. 1) have been studied using piezoelectric quartz-crystal resonators. In most cases, the resonator vibrates in a thickness-shear mode and generates a shear wave in the liquid. Due to viscous damping of the liquid, the shear wave decays very fast over a short distance, which leads to a shift in the frequency of the resonator and a drop in its quality factor.

Since mid-1990s, some experiments have shown that the frequency shift for a thickness-shear mode resonator in liquid actually depends on the boundary condition and oscillates with the thickness of a finite liquid layer (Refs. 2 and 3). Theoretical models have been put forward, including a piece of my own work (Ref. 4), which explains the oscillation as a result of the propagation of compressional wave in the liquid and its reflection at the liquid surface or interface. It is found that a compressional wave can propagate a much longer distance in a liquid than the shear wave, despite the viscous damping. This poses a challenge for the shear-mode resonators as liquid-phase sensors because thickness-shear mode vibrations are almost always coupled with a flexural mode that generates a compressional wave in the liquid.

For the microcantilever resonator, I believe it operates in a flexural mode and thus the effect of the compressional wave in the liquid must be carefully examined. For example, does the resonator frequency or Q-factor depend on the size of the liquid cell?

Another question I have is: Is Q = 15 really a high quality factor? Recently we reported a quality factor of 7000 for a torsional-mode resonator in water (Ref. 5). Of course, the Q would be lower for a higher viscosity liquid.

Finally, I was curious about what piezoelectric material was used in this study, but could not find the information in the paper.

Some references:

Kanazawa and Gordon, Anal. Chim. Acta. 175, 99-105 (1985).
Schneider and Martin, Anal. Chem. 67, 3324-3335 (1995).
Lin and Ward, Anal. Chem. 67, 685-693 (1995).
P.C.Y. Lee and R. Huang, Effects of a liquid layer on thickness-shear vibrations of rectangular AT-cut quartz plates. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 49, 604-611 (2002).
T. Knowles, M.K. Kang, R. Huang, Trapped torsional vibrations in elastic plates. Applied Physics Letters 87, 201911 (2005).

Lu's paper - some explanation

Stanislaw Kalicinski — Thu, 22 Mar 2007 09:17:43 -0400

Hi,

Can anybody help me with understanding of some terms from Lu's paper? It seems I lack of some principle knowledge about physics at surfaces. My background is mostly micro-electronics and MEMS but I'm doing PhD in reliability of MEMS and would like to understand more what is happening on the surface of semiconductors (from mechanical point of view) in case of adsorption-desorption of different spieces.

I do not understand what is strain-dependent and independent surface stress, the two terms mentioned in the paper mentioned by Mr. Pradeep.

What is still not clear to me is whether the surface stress affect the resonance frequency of cantilevers or not. It seems that what Lagowski have written is not true (see Gurtin et al) but Lu says it true in one case but not in another (strain-dependent surface stress). This is very confusing.

Could you indicate what I should read (books, papers) first to understand it?

Thank you in advance!

ask for a help

Zaiwang Huang — Mon, 18 Dec 2006 21:36:11 -0500

Dear weixu your result is really interesting for me. for my current research, I encounter a problem similar to you. in my experiment of uniaxial compression for micropillar, the elastic modulus of different diameter(micro to submicro-scale) column vary largely, could you provide me some your research details? thanks a lot

Thank you I have posted it!

Weixu Zhang — Mon, 18 Dec 2006 21:17:54 -0500

Thank you Prof. Sharma I have posted the paper on the effect of surface energy on yield strength!

You can (nearly) always post a pre-print.

Pradeep Sharma — Mon, 18 Dec 2006 19:51:24 -0500

You can always post a pre-print. Far as I know, this is not a violation of copyright. You can check rules for each journal on this web site.

To add an attachment, you have to create your own new post which I would encourage you to do.

I am glad to post my manuscript here, but I don't know copyrigh

Weixu Zhang — Mon, 18 Dec 2006 19:32:18 -0500

Prof. Sharma.
Yes I am glad to post my manuscript here, and I think it may be helpful to understand the surface effect more.

But I know nothing about the copyright problem. It is just a manuscript submited to APL under review. Can I post it here ? Of course it is my own manuscript. Another question is that if I have the rights to allow others to download my papers published in a journal?
It is part of my doctoral dissertation. If I can post it here, who will please tell me how to add an attachment here?

Please do post it...

Pradeep Sharma — Mon, 18 Dec 2006 11:10:38 -0500

Weixu, please do post your paper here. Along with me, I am sure, others will be interested to look at it.

Hi Prof. Sharma would like to see my manuscript on surface effe

Weixu Zhang — Mon, 18 Dec 2006 10:28:45 -0500

Hi Prof. Sharma would like to see my manuscript on surface effect on the yield strength of material with nano-inhomogeneities?

I investigated the surface effect on plastic deformation of nanomaterials. The surface effect has influence not only on the elastic moduli but also plastic deformation of materials with nano-inhomogeneities.