Vibration boundary control of Timoshenko micro-cantilever using piezoelectric actuators

Document Type : Article

Authors

1 Department of Mechanical Engineering, Sharif University of Technology, Azadi Ave, Tehran, Iran

2 School of Mechanical Engineering, Shiraz University, Shiraz, Iran

Abstract

One of the methods of force/moment exertion to micro beams is utilizing piezoelectric actuators. In this paper the problem of vibration boundary control of a clamped-free Timoshenko micro-cantilever considering effects of the piezoelectric actuator to achieve asymptotic stability is addressed. To achieve this purpose, the dynamic equations of the beam actuated by a piezoelectric layer laminated on one side of the beam are extracted. The control law has been made to decay vibrations of the beam. This control law is achieved from the feedback of time derivatives of boundary states of the beam. The obtained control is applied in the form of voltage of the piezoelectric. To illustrate the impact of the proposed controller to the micro beam, the finite element method and Timoshenko beam element have been used and then the simulation has been performed. The simulation shows that not only does this control voltage decay the vibration of the beam, but also the mathematical proofs proposed in this article are precise and implementable. 

Keywords

Main Subjects

References

References

1. Indermuhle, P., Schurmann, G., Racine, G., and De Rooij, N. \Atomic force microscopy using cantilevers with integrated tips and piezoelectric layers for actuation and detection", Journal of Micromechanics and Microengineering, 7(3), p. 218 (1997).
2. Maluf, N. and Williams, K., Introduction to Microelectromechanical
Systems Engineering, Artech House
(2004).
3. Zhang, W., Meng, G., and Li, H. \Adaptive vibration
control of micro-cantilever beam with piezoelectric
actuator in MEMS", The International Journal of
Advanced Manufacturing Technology, 28(3-4), pp. 321-
327 (2006).
4. Gahlin, R. and Jacobson, S. \A novel method to map
and quantify wear on a micro-scale", Wear, 222(2),
pp. 93-102 (1998).
5. Garca, R., Calleja, M., and Perez-Murano, F. \Local
oxidation of silicon surfaces by dynamic force microscopy:
Nanofabrication and water bridge formation",
Applied Physics Letters, 72(18), pp. 2295-2297
(1998).
6. Miyahara, K., Nagashima, N., Ohmura, T., and
Matsuoka, S. \Evaluation of mechanical properties
in nanometer scale using AFM-based nanoindentation
tester", Nanostructured Materials, 12(5), pp. 1049-
1052 (1999).
7. Furlani, E. \Simulation of grating light valves", in
Technical Proceeding of the 1998 International Conference
on Modeling and Simulation of Microsystems
(1998).
8. Arik, M., Zurn, S., Bar-Cohen, A., Nam, Y., Markus,
D., and Polla, D. \Development of CAD model for
MEMS micropumps", in Technical Proceedings of the
1999 International Conference on Modeling and Simulation
of Microsystems (1999).
9. Bernstein, D., Guidotti, P., and Pelesko, J. \Mathematical
analysis of an electrostatically actuated MEMS
device", Proceedings of the Modeling and Simulation of
Microsystems MSM, pp. 489-492 (2000).
10. Liu, J., Mei, Y., Xia, R., and Zhu, W. \Large
displacement of a static bending nanowire with surface
e ects", Physica E: Low-Dimensional Systems and
Nanostructures, 44(10), pp. 2050-2055 (2012).
11. Alhazza, K.A., Daqaq, M.F., Nayfeh, A.H., and
Inman, D.J. \Non-linear vibrations of parametrically
excited cantilever beams subjected to non-linear
A. Mehrvarz et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 711{720 719
delayed-feedback control", International Journal of
Non-Linear Mechanics, 43(8), pp. 801-812 (2008).
12. Zhao, D., Liu, J., and Wang, L. \Nonlinear free
vibration of a cantilever nanobeam with surface e ects:
Semi-analytical solutions", International Journal of
Mechanical Sciences, 113, pp. 184-195 (2016).
13. McCarthy, B., Adams, G.G., McGruer, N.E., and Potter,
D. \A dynamic model, including contact bounce,
of an electrostatically actuated microswitch", Journal
of, Microelectromechanical Systems, 11(3), pp. 276-283
(2002).
14. Jalili, N. and Laxminarayana, K. \A review of atomic
force microscopy imaging systems: application to
molecular metrology and biological sciences", Mechatronics,
14(8), pp. 907-945 (2004).
15. Krstic, M., Guo, B.-Z., Balogh, A., and Smyshlyaev,
A. \Control of a tip-force destabilized shear beam by
observer-based boundary feedback", SIAM Journal on
Control and Optimization, 47(2), pp. 553-574 (2008).
16. Shirazi, M.J., Salarieh, H., Alasty, A., and Shabani, R.
\Tip tracking control of a micro-cantilever Timoshenko
beam via piezoelectric actuator", Journal of Vibration
and Control, 19(10), pp. 1561-1574 (2013).
17. Canbolat, H., Dawson, D., Rahn, C., and Vedagarbha,
P. \Boundary control of a cantilevered
exible beam
with point-mass dynamics at the free end", Mechatronics,
8(2), pp. 163-186 (1998).
18. Dogan, M. and Morgul, O. \Boundary control of a rotating
shear beam with observer feedback", Journal of
Vibration and Control, 18(14), pp. 2257-2265 (2011).
19. Fard, M. and Sagatun, S. \Exponential stabilization of
a transversely vibrating beam via boundary control",
Journal of Sound and Vibration, 240(4), pp. 613-622
(2001).
20. Sadek, I., Kucuk, I., Zeini, E., and Adali, S. \Optimal
boundary control of dynamics responses of piezo actuating
micro-beams", Applied Mathematical Modelling,
33(8), pp. 3343-3353 (2009).
21. Vatankhah, R., Naja , A., Salarieh, H., and Alasty, A.
\Asymptotic decay rate of non-classical strain gradient
Timoshenko micro-cantilevers by boundary feedback",
Journal of Mechanical Science and Technology, 28(2),
pp. 627-635 (2014).
22. He, W., Ge, S.S., How, B.V.E., Choo, Y.S., and Hong,
K.-S. \Robust adaptive boundary control of a
exible
marine riser with vessel dynamics", Automatica, 47(4),
pp. 722-732 (2011).
23. Yang, K.-J., Hong, K.-S., and Matsuno, F. \Boundary
control of a translating tensioned beam with varying
speed", Mechatronics, IEEE/ASME Transactions on,
10(5), pp. 594-597 (2005).
24. Vatankhah, R., Naja , A., Salarieh, H., and Alasty,
A. \Boundary stabilization of non-classical micro-scale
beams", Applied Mathematical Modelling, 37(20), pp.
8709-8724 (2013).
25. Vatankhah, R., Naja , A., Salarieh, H., and Alasty,
A. \Exact boundary controllability of vibrating nonclassical
Euler-Bernoulli micro-scale beams", Journal
of Mathematical Analysis and Applications, 418(2),
pp. 985-997 (2014).
26. He, W., Ge, S.S., and Zhang, S. \Adaptive boundary
control of a
exible marine installation system", Automatica,
47(12), pp. 2728-2734 (2011).
27. How, B., Ge, S., and Choo, Y. \Active control of
exible
marine risers", Journal of Sound and Vibration,
320(4), pp. 758-776 (2009).
28. Nguyen, T., Do, K.D., and Pan, J. \Boundary control
of coupled nonlinear three dimensional marine risers",
Journal of Marine Science and Application, 12(1), pp.
72-88 (2013).
29. Paranjape, A.A., Guan, J., Chung, S.-J., and Krstic,
M. \PDE boundary control for
exible articulated
wings on a robotic aircraft", Robotics, IEEE Transactions
on, 29(3), pp. 625-640 (2013).
30. Han, S.M., Benaroya, H., and Wei, T. \Dynamics of
transversely vibrating beams using four engineering
theories", Journal of Sound and Vibration, 225(5), pp.
935-988 (1999).
31. Reddy, J.N., Applied Functional Analysis and Variational
Methods in Engineering, McGraw-Hill College
(1986).
32. Robinson, J.C., In nite-Dimensional Dynamical Systems:
An Introduction to Dissipative Parabolic PDEs
and the Theory of Global Attractors, 28, Cambridge
University Press (2001).
33. Yosida, K., Functional Analysis, Springer (1980).
34. Pazy, A., Semigroups of Linear Operators and Applications
to Partial Di erential Equations, 44, Springer
New York (1983).
35. Guo, B.-Z. and Morgul,  O., Stability and Stabilization
of In nite Dimensional Systems with Applications,
Springer Science & Business Media (1999).
36. Huebner, K.H., Dewhirst, D.L., Smith, D.E., and Byrom,
T.G., The Finite Element Method for Engineers,
John Wiley & Sons (2008).
37. Gad-el-Hak, M., MEMS: Introduction and Fundamentals,
CRC press (2005).


Scientia Iranica
Volume 25, Issue 2
Transactions on Mechanical Engineering (B)
March and April 2018
Pages 711-720

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History

  • Receive Date: 27 April 2016
  • Revise Date: 03 January 2017
  • Accept Date: 17 April 2017

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