The problem of forced monoharmonic, axisymmetric, bending vibrations and dissipative heating of circular viscoelastic plates with piezoelectric sensors and actuators is considered. We describe the viscoelastic behavior of a passive (without the piezoeffect) and a piezoactive material according to the concept of complex modules depending on temperature. The nonlinear coupled problem of electrothermoviscoelasticity is solved by numerical methods. The influence of boundary conditions and temperature of dissipative heating on the active damping of forced resonant vibrations of circular viscoelastic plates using piezoelectric sensors and actuators is investigated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. K. Godunov, “On the numerical solution of boundary-value problems for systems of ordinary differential equations,” Usp. Mat. Nauk, 16, No. 3, 171–174 (1961).
Ya. M. Hryhorenko and A. P. Mukoed, Solution of Linear and Nonlinear Problems of the Theory of Shells on Computers [in Ukrainian], Lybid’, Kyiv (1992).
V. G. Karnaukhov, T. V. Karnaukhova, and O. V. P’yatets’ka, “Influence of the temperature of dissipative heating on the active damping of forced axisymmetric vibrations of a circular plate using a piezoelectric actuator,” Visn. Kyiv. Univ. Ser. Fiz.-Mat. Nauky, Issue 3, 107–114 (2004).
V. G. Karnaukhov, T. V. Karnaukhova, and O. V. P’yatets’ka, “Influence of the temperature of dissipative heating on indices of a piezoelectric sensor under forced axisymmetric vibrations of a circular plate,” Visn. Kyiv. Univ. Ser. Fiz.-Mat. Nauky, Issue 1, 107–113 (2004).
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, V. I. Kozlov, and T. V. Karnaukhova, “Refined thermomechanical Timoshenko-type model of composite shells with distributed transversally isotropic sensors under monoharmonic loading,” Mat. Met. Fiz.-Mekh. Polya, 49, No. 4, 119–132 (2006).
V. G. Karnaukhov, V. I. Kozlov, and T. V. Karnaukhova, “Refined thermomechanical Timoshenko-type model of composite shells with distributed transversally isotropic actuators under monoharmonic loading,” Prykl. Probl. Mekh. Mat., Issue 4, 84–95 (2006).
V. G. Karnaukhov, V. I. Kozlov, Ya. O. Zhuk, and T. V. Karnaukhova, “Thermomechanical coupled theory of layered shells with passive, physically nonlinear, inelastic layers and distributed piezoelectric inclusions for control of nonstationary vibrations,” Mat. Met. Fiz.-Mekh. Polya, 44, No. 3, 96–106 (2001).
V. G. Karnaukhov and V. V. Mikhailenko, Nonlinear Thermomechanics of Piezoelectric Inelastic Bodies under Monoharmonic Loading [in Russian], Zhitomir Engineering–Technological Institute, Zhitomir (2005).
V. G. Karnaukhov, A. V. Kozlov, and E. V. Pyatetskaya, “Active damping of axisymmetric vibrations of a circular plate using distributed sensors and actuators,” in: System Technologies. Issue “Mathematical Problems of Industrial Mechanics,” [in Ukrainian], Dnipropetrovs’k (2003), pp. 145–151.
V. G. Karnaukhov, V. I. Kozlov, and E. V. Pyatetskaya, “Active damping of vibrations of a rectangular plate using distributed sensors and actuators,” Teor. Prikl. Mekh., Issue. 37, 137–140 (2003).
V. G. Karnaukhov, V. I. Kozlov, and O. V. P’yatets’ka, “Active damping of axisymmetric vibrations of a circular plate using piezoelectric inclusions,” Visn. Kyiv. Univ. Ser. Fiz.-Mat. Nauky, Issue 2, 81–85 (2003).
I. F. Kirichok, “Flexural vibrations and vibrational heating of a ring plate with thin piezoceramic covers under single-frequency electromechanical loading,” Prikl. Mekh., 44, No. 2, 99–108 (2008), English translation: Int. Appl. Mech., 44, No. 2, 200–207 (2008).
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 M. V. Karnaukhov, “Monoharmonic vibrations and vibrational heating of an electromechanically loaded circular plate with piezoelectric actuators subject to shear strain,” Prikl. Mekh., 44, No. 9, 104–114 (2008), English translation: Int. Appl. Mech., 44, No. 9, 1041–1049 (2008).
I. F. Kirichok and M. V. Karnaukhov, “Single-frequency vibrations and vibrational heating of a piezoelectric circular sandwich plate under monoharmonic electromechanical loading,” Prikl. Mekh., 44, No. 1, 79–89 (2008), English translation: Int. Appl. Mech., 44, No. 1, 65–72 (2008).
I. F. Kyrychok, O. V. P’yatets’ka, and M. V. Karnaukhov, “Bending vibrations and dissipative heating of a circular viscoelastic plate with piezoelectric actuators under electromechanical monoharmonic loading,” Visn. Kyiv. Univ. Ser. Fiz.-Mat. Nauky, Issue 2, 84–92 (2006).
V. V. Matveev, Damping of Vibrations of Deformable Bodies [in Russian], Naukova Dumka, Kiev (1985).
A. D. Nashif, D. I. G. Jones, and J. P. Henderson, Vibration Damping, Wiley, New York (1985).
N. P. Nesterenko, O. P. Chervinko, and I. K. Senchenkov, “Modeling of ultrasonic heating of structural elements of polymeric fibrous composites under normal loading,” Vest. Nats. Tekhn. Univ. “KhPI,” No. 9, 3–8 (2002).
N. A. Shul’ga and A. M. Bolkisev, Vibrations of Piezoelectric Bodies [in Russian], Naukova Dumka, Kiev (1990).
U. Gabbert and H. S. Tzou, Smart Structures and Structronic Systems, Kluwer, Dordrecht (2001).
V. G. Karnaukhov, I. F. Kirichok, and M. V. Karnaukhov, “The influence of dissipative heating on active vibration damping of viscoelastic plates,” J. Eng. Math., 61, Nos. 2–4, 399–411 (2008).
H. S. Tzou, Piezoelectric Shells (Distributed Sensing and Control of Continua), Kluwer, Dordrecht (1993).
H. S. Tzou and L. A. Bergman, Dynamics and Control of Distributed Systems, Cambridge University Press, Cambridge (1998).
Author information
Authors and Affiliations
Additional information
Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 53, No. 2, pp. 94–107, April–June, 2010.
Rights and permissions
About this article
Cite this article
Kyrychok, I.F., Karnaukhova, T.V. 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. J Math Sci 178, 480–495 (2011). https://doi.org/10.1007/s10958-011-0564-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-011-0564-7