Journal of Mathematical Sciences

, Volume 184, Issue 1, pp 78–87 | Cite as

Active control of the strained state of an asymmetric trimorphic beam under nonstationary modes of operation

  • A. E. Babaev
  • I. V. Yanchevskiy

We consider the problem of active control of the strained state of an asymmetric trimorphic beam with floating end faces. The nonstationary mechanical load applied to the trimorph is an unknown function of time. The electric signal, varying in time according to a law that is determined by the proposed criterion, is supplied to the electrodes of one of the piezolayers working under the mode of inverse piezoelectric effect. Here, the state of the beam is sustained close to unstrained. The formation of the controlling electric signal and the identification of the external mechanical action are carried out on the basis of the known potential difference, which arises between the electrodes of the second piezolayer, working under the mode of direct piezoelectric effect. The problem is solved using the Laplace integral transformation in time. As a result of the analytic transition to the space of originals, we determine the required quantities from the system of Volterra integral equations, which is solved with the help of regularizing algorithms. The results of calculations and their analysis are also presented.


Inverse Piezoelectric Effect Direct Piezoelectric Effect Nonstationary Vibration Potential Difference Versus Bimorph Beam 


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  1. 1.
    A. E. Babaev, A. A. Babaev, and I. V. Yanchevskiy, “Nonstationary vibrations of a bimorph beam under the modes of direct and inverse piezoelectric effect,” Aktualn. Probl. Fiz.-Mekh. Issled. Akust. Volny, No. 3, 16–27 (2007).Google Scholar
  2. 2.
    A. E. Babaev and Yu. B. Moseenkov, “Nonstationary vibrations of a thin-walled electroelastic band,” Dopov. Akad. Nauk Ukrainy, No. 12, 54–58 (1994).Google Scholar
  3. 3.
    A. N. Guz’ (editor), Mechanics of Coupled Fields in Structural Elements [in Russian], Vol. 5: V. T. Grinchenko, A. F. Ulitko, and N. A. Shul’ga, Electroelasticity, Naukova Dumka, Kiev (1989).Google Scholar
  4. 4.
    V. A. Ditkin and A. P. Prudnikov, Handbook on Operational Calculus [in Russian], Vysshaya Shkola, Moscow (1965).Google Scholar
  5. 5.
    V. S. Didkovs’kyi, O. G. Leiko, and V. G. Savin, Electroacoustic Piezoelectric Transducers (Calculation, Projection, and Construction) [in Ukrainian], Imeks-LTD, Kirovohrad (2006).Google Scholar
  6. 6.
    V. G. Karnaukhov and T. V. Karnaukhova, “Damping of the resonance flexural vibrations of a flexible hinged viscoelastic round plate in the case of joint use of sensors and actuators,” Teor. Prikl. Mekh., No. 46, 125–131 (2009).Google Scholar
  7. 7.
    V. M. Sharapov, M. P. Musienko, and E. V. Sharapova, Piezoelectric Transducers of Physical Quantities [in Russian], Cherkassy Gos. Tekhnol. Univ., Cherkassy (2005).Google Scholar
  8. 8.
    I. V. Yanchevskiy, “Identification of a nonstationary load acting on an asymmetric bimorph,” in: Bulletin of the “KhPI” National Technical University, Dynamics and Strength of Machines [in Russian], Issue 36 (2008), pp. 184–190.Google Scholar
  9. 9.
    E. G. Yanyutin, I. V. Yanchevskiy, A. V. Voropai, and A. S. Sharapata, Problems of the Impulse Deformation of Structural Elements [in Russian], Izd. Kharkov Nats. Avtodor. Univ., Kharkov (2004).Google Scholar
  10. 10.
    M. Collet, V. Walter, and P. Delobelle, “Active damping of a micro-cantilever piezo-composite beam,” J. Sound Vibr., 260, No. 3, 453–476 (2003).MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    A. Donoso and O. Sigmund, “Optimization of piezoelectric bimorph actuators with active damping for static and dynamic loads,” Struct. Multidisc Optim., 38, 171–183 (2009).CrossRefGoogle Scholar
  12. 12.
    Z. Gosiewski and Z. P. Koszewnik, “Modeling of beam as control plane for a vibration control system,” Solid State Phenomena, 144, Mechatronic Systems and Materials II, 59–64 (2009).Google Scholar
  13. 13.
    T. V. Karnaukhova, “Active damping of forced resonance vibrations of an isotropic shallow viscoelastic cylindrical panel under the action of an unknown mechanical load,” Mat. Metody Fiz.-Mekh. Polya, 52, No. 1, 84–91 (2009); English translation: J. Math. Sci., 168, Nо. 4, 603–612 (2009).CrossRefGoogle Scholar
  14. 14.
    S. I. Rudnitskii, V. M. Sharapov, and N. A. Shul’ga, “Vibrations of a bimorphic disk transducer of the metal-piezoceramic type,” Prikl. Mekh., 26, No. 10, 64–72 (1990); English translation: Int. Appl. Mech., 26, No. 10, 973–980 (1990).Google Scholar
  15. 15.
    J. Tani, T. Takagi, and J. Qiu, “Intelligent material systems: application of functional materials,” Appl. Mech. Rev., No. 51, 505–521 (1998).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  • A. E. Babaev
    • 1
  • I. V. Yanchevskiy
    • 1
  1. 1.KharkivUkraine

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