• I. F. Kyrychok S. P. Timoshenko Institute of Mechanics
  • Yа. O. Zhuk Taras Shevchenko National University of Kyiv
  • O. A. Chernyushok National University of Food Technologies
  • A. P. Tarasov Taras Shevchenko National University of Kyiv
Keywords: resonance vibration, dissipative heating, inelastic material, piezoelectric sensors


The problem of forced resonance oscillations and dissociative heating of a hinged cylindrical shell with a piezoelectric sensor is considered. The formulation of the problem is carried out within the framework of the refined theory of shells by S.P. Tymoshenko. The object of study is a three-layer cylindrical shell loaded with axisymmetric surface pressure. The external load changes in time according to the harmonic law. The inner layer of the shell is made of passive transversal-isotropic material. The outer layers are made of sandstone, polarized in thickness in oppo-site directions. The materials of the layers are viscoelastic and their properties are temperature dependent. The surfaces in contact with the passive layer are covered with continuous, infinitesimally thin electrodes on which the electric potential is given. The problem of electro-thermomechanics under consideration is interconnected and non-linear. For this problem, physical, geometric and motion equations are obtained. The method of its solving is constructed and implemented. The problem of forced oscillation and vibration of the shell is reduced to the solution of the partial differential equation system with known boundary and initial conditions. A step-by-step method is used to integrate it. The numerical solution of the problem is given. The calculations were performed in the vicinity of the first frequency of the bending mode oscillations of the shell. The obtained numerical solution is analyzed.


1. Bolkisev A. M., Karlash V. L., Shul’ga N. A. Temperature dependence of the properties of piezoelectric ceramics. Sov. Appl. Mech. 1984. Vol. 20, No 7. P. 650–653.
2. Карнаухов В. Г., Киричок И. Ф. Связанные задачи теории вязкоупругих пластин и оболочек. Киев: Наук. думка, 1986. 222 с.
3. Карнаухов В. Г., Михайленко В. В. Нелинейная термомеханики пьезоэлектрических неупругих тел при моногармоническом нагружении. Житомир: ЖГТУ, 2005. 428 с.
4. Киричок И. Ф. Осесимметричные резонансные колебания и разогрев оболочек вращения и их контроль пьезоэлектрическими сенсорами и актуаторами. Прикл. механика. 2010. Т. 46, № 8. С. 42–57.
5. Киричок И. Ф., Жук Я. А. Влияние граничных условий и температуры виброразогрева на резонансные осесимметричные колебания вязкоупругих цилиндрических оболочек с пьезоактуаторами и сенсорами. Теоретическая и прикладная механика. 2013. Вып. 7. С. 133–140.
6. Киричок І. Ф., Жук Я. О., Карнаухова Т. В. Резонансні коливання і дисипативний розігрів гнучкої в’язкопружної балки з п’єзоактуаторами при врахуванні деформації зсуву. Мат. методи та фіз.-мех. поля. 2017. Т. 60, № 2. С. 66–74.
7. Киричок И. Ф., Карнаухова Т. В., Пересунько Н. В. Резонансные осесимметричные колебания и диссипативный разогрев цилиндрических оболочек и их контроль с помощью пьезоэлектрических актуаторов. Теоретическая и прикладная механика. 2010. Вып. 46. С. 132–140.
8. Blaguenon A., Lene F., Bemadou М. Active control a beam a piezoceramic element. Smart Mater. Struct. 1999. No 8. P. 116–124.
9. Gabbert U., Tzou H. S. Smart structures and structronic systems. Dordrecht: Kluver acad. publ., 2001. 384 p.
10. Karnaukhov V. G., Kyrychok I. F., Kozlov V. I. Thermomechanics of Inelastic Thin-Wall Structural Members with Piezoelectric Sensors and Actuators under Harmonic Loading (Review). Int. App. Mech. 2017. Vol. 53, Issue l. P. 6–58.
11. Karnaukhov V. G., Kozlov V. I., Karnaukhova T. V. Forced Vibrations and Dissipative Heating of flexible Hinged Viscoelastic Rectangular Plates with Actuators with Allowance for Transverse Shear Strains. Int. Appl. Mech. 2018. Vol. 54, Issue 1. P. 85–93.
12. Kyrychok I. F. Resonance Axisymmetric Vibrations and Dissipative Heating of Viscoelastic Cylindrical Shell with Piezolayers under Elektromechanical Excitation. Int. Appl. Mech. 2015. Vol. 51, Issue 5. P. 567–573.
13. Kyrychok I. F., Chemyshok O. A. Forced Vibrations and Vibroheating of flexible Viscoelastic Beam with Piezoelectric Sensor and Actuators with Allowance for the Shear Strains. Int. Appl. Mech. 2018. Volume 54, Issue 5. P. 568–576.
14. Tani J., Takagia Т., Qiu J. Intelligent material systems: Application of functional materials. Appl. Mech. Review. 1998. Vol. 51, Issue 8. P. 505–521.
15. Tzou H. S. Piezoelectric Shells (Distributed Sensing and Control of Continua). Dordrecht–Boston–London: Kluwer Academic Publishers, 1993. 400 p.
16. Tzou H. S., Bergman L. A. Dynamics and Control of Distributed Systems. Cambridge: Cam-bridge University Pres, 1998. 374 p.
How to Cite
Kyrychok, I. F., Zhuk, Y. O., Chernyushok, O. A., & Tarasov, A. P. (2020). FORMULATION OF THE PROBLEMS ON RESONANCE VIBRA-TIONS AND DISPIPATIVE HEATING OF THE FLEXIBLE CYLIN-DRICAL SHEETS WITH PIEZOACTIVE BALLS TAKING INTO CONSIDERATION OF SHIFT DEFORMATION. Bulletin of Zaporizhzhia National University. Physical and Mathematical Sciences, (2), 55-61. Retrieved from