We consider the problem of the forced resonance vibrations and dissipative heating of a viscoelastic beam with piezoelectric actuators with regard for geometric nonlinearity in the quadratic approximation. The viscoelastic behavior of passive (without the piezoelectric effect) and piezoactive materials is described in terms of instantaneous and complex moduli. To solve the nonlinear problem of electroviscoelasticity and heat conduction, we use the method of quasilinearization together with the numerical methods of discrete orthogonalization and finite differences. We have studied the influence of geometric nonlinearity on the dynamic characteristics, temperature of vibration heating, and active damping of a beam with using a piezoelectric actuator.
Similar content being viewed by others
References
A. M. Bolkisev, V. L. Karlash, and N. A. Shul’ga, “Temperature dependence of the properties of piezoelectric ceramics,” Prikl. Mekh., 20, No. 7, 70–74 (1984); English translation: Int. Appl. Mech., 20, No. 7, 650–653 (1984).
M. V. Vasylenko and O. M. Alekseichuk, Theory of Oscillations and Stability of Motion [in Ukrainian], Vyshcha Shkola, Kyiv (2004).
Ya. M. Grigorenko, G. G. Vlaikov, and A. Ya. Grigorenko, Numerical and Analytical Solution of the Problems of Shell Mechanics Based on Different Models [in Russian], Akademperiodika, Kiev (2006).
Ya. A. Zhuk and I. K. Senchenkov, “Modeling the stationary vibrations and dissipative heating of thin-walled inelastic elements with piezoactive layers,” Prikl. Mekh., 40, No. 5, 80–91 (2004); English translation: Int. Appl. Mech., 40, No. 5, 546–556 (2004).
A. A. Il’yushin and B. E. Pobedrya, Fundamentals of the Mathematical Theory of Thermoviscoelasticity [in Russian], Nauka, Moscow (1970).
V. G. Karnaukhov and I. F. Kirichok, Coupled Problems of the Theory of Viscoelastic Plates and Shells [in Russian], Naukova Dumka, Kiev (1986).
V. G. Karnaukhov and I. F. Kirichok, Electrothermoviscoelasticity [in Russian], Naukova Dumka, Kiev (1988).
V. G. Karnaukhov, I. F. Kirichok, and V. I. Kozlov, “Effect of the temperature of dissipative heating on the active damping of forced vibrations of inelastic thin plates with the help of piezoelectric sensors and actuators,” in: Topical Aspects of Physicomechanical Investigations. Acoustics and Waves [in Ukrainian], Naukova Dumka, Kyiv (2007), pp. 127–152.
V. G. Karnaukhov, I. F. Kirichok, and V. I. Kozlov, “Electromechanical oscillations of thin-walled elements with piezoelectric effect,” in: Advances in Mechanics [in Russian], Vol. 2, A.S.K., Kiev (2006), pp. 185–217.
V. G. Karnaukhov, V. I. Kozlov, and E. V. Pyatetskaya, “Active damping of the vibrations of a rectangular plate with the help of distributed sensors and actuators,” in: Theoretical and Applied Mechanics [in Russian] (2003), pp. 137–140.
V. G. Karnaukhov, V. I. Kozlov, and T. V. Karnaukhova, “Modeling of the forced resonance vibrations and dissipative heating of flexible viscoelastic plates with distributed actuators,” in: Physical-and-Mathematical Modeling and Information Technologies [in Ukrainian], Issue 8 (2008), pp. 48–68.
V. G. Karnaukhov and V. V. Mikhailenko, Nonlinear Thermomechanics of Piezoelectric Inelastic Bodies under Monoharmonic Loading [in Russian], Zhitomir State Technological University, Zhitomir (2005).
I. F. Kirichok, “Resonant vibration and heating of ring plates with piezoactuators under electromechanical loading and shear deformation,” Prikl. Mekh., 45, No. 2, 124–132 (2009); English translation: Int. Appl. Mech., 45, No. 2, 215–222 (2009).
I. F. Kirichok and T. V. Karnaukhova, “Influence of boundary conditions and temperature of dissipative heating on active damping of forced axisymmetric resonant bending vibrations of circular viscoelastic plates by piezoelectric sensors and actuators,” Mat. Metody Fiz.-Mekh. Polya, 53, No. 2, 94–107 (2010); English translation: J. Math. Sci., 178, No. 5, 480–495 (2011).
I. F. Kirichok and T. V. Karnaukhova, “Control of forced vibrations of round viscoelastic plates with the help of piezoelectric sensors and actuators with regard for vibration heating,” in: Physical-and-Mathematical Modeling and Information Technologies [in Ukrainian], Issue 9 (2009), pp. 67–78.
I. K. Senchenkov and I. F. Kirichok, “Forced nonlinear vibrations and dissipative heating of a viscoelastic beam,” Prikl. Mekh., 23, No. 1, 91–97 (1987); English translation: Int. Appl. Mech., 23, No. 1, 80–86 (1987).
K. K. Stevens, “Transverse vibration of a viscoelastic column with initial curvature under periodic axial load,” Trudy Am. Obshch. Inzh.-Mekh., Ser. E: Prikl. Mekh., No. 4, 168–173 (1969).
A. Blаguenon, F. Lene, and M. Bernadou, “Active control of a beam using a piezoceramic element,” Smart Mater. Struct., 8, 116–124 (1999).
M. Brennan, S. Elliot, and R. Pinnington, “The dynamic coupling between piezoceramic actuators and a beam,” J. Acoust. Soc. Amer., 102, No. 4, 1931–1942 (1997).
U. Gabbert and H. S. Tzou, Smart Structures and Structronic Systems, Kluwer, Dordrecht (2001).
H. S. Tzou and G. L. Anderson (editors), Intelligent Structural Systems, Kluwer, Dordrecht (1992).
V. G. Karnaukhov, I. F. Kirichoк, and M. V. Karnaukhov, “The influence of dissipative heating on active vibration damping of viscoelastic plates,” J. Eng. Math., 61, No. 2–4, 399–411 (2008).
H. S. Tzou, Piezoelectric Shells (Distributed Sensing and Control of Continua), Kluwer, Dordrecht (1993).
Author information
Authors and Affiliations
Additional information
Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 54, No. 4, pp. 120–130, October–December, 2011.
Rights and permissions
About this article
Cite this article
Kirichok, I.F., Senchenkov, I.K. & Chervinko, O.P. Forced resonance vibrations and dissipative heating of a flexible viscoelastic beam with piezoelectric actuators. J Math Sci 187, 685–698 (2012). https://doi.org/10.1007/s10958-012-1093-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-012-1093-8