Nonlinear oscillations of elastic beam including dissipation and the galerlkin method in their investigation

Kuzio I., Pukach P., Nytrebych Z., Sokhan P.

Department of Mechanics and Automation Engineering Lviv Polytecnic National University

The paper is devoted to the research of the oscillating system that is described by the first mixed problem for the weakly nonlinear equation of the  beam vibrations in a bounded domain. The conditions of the existence of the local, according to a time variable, solution have been obtained. Oscillating blowup regime is especially highlighted.  The possibility of the Galerkin method application to the problem is shown.

1. Gu R.J., Kuttler K.L., Shillor M. Frictional wear of a thermoelastic beam // J. Math. Anal. And Appl. – 242. – 2000. – P. 212 – 236. 2. Erofeev V.I., Kazhayev V.V., Semerikova N.P. Volny v sterzhnyakh. Dispersiya. Dissipaciya. Nelineynost [Waves in rods. Dispersion. Dissipation. Non-linearity].–Мoscow: FIZMATLIT, 2002. – 208 p.  3. Sokil B.I., Senyk А.P., Nazar I.I., Sokil M.B. Poperechni kolyvannia neliniyno pruzhnogo seredovyshcha i metod D’Alambera u yikh doslidzhenni [Transversal vibrations of nonlinear elastic environment and method of D’Alambert in their research] // Visn. DU "Lviv. politekhnika" [Herald of "Lviv. polytechnics" University]. – 2004. – No. 509. – pp. 105–110. 4. Pukach P.Ya. Zmishana zadacha dlya odnoho sylno neliniynoho rivnyannya typu kolyvan balky v obmezheniy oblasti [Mixed problem for some strongly nonlinear equation of beam vibrations type in bounded domain] // Prykladni problemy mekhaniky ta matematyky [Applied problems of mechanics and mathematics]. –2006. – Issue 4.–Pages 59 –69. 5. Pukach P., Kuzio I., Sokil M. Qualitative methods for re¬search of trans¬versal vibrations of semi-infinite cable under the action of nonlinear resistance forces // ECONTECHMOD. - 2013.- Volume 2, Is. 1. - Pages 43-48. 6. Buhrii О., Domanska H., Protsakh N. Zmishana zadacha dlya neliniynoho rivnyannya tretioho poryadku v uzahalnenykh prostorakh Soboleva [Mixed problem for nonlinear equation of third order in Sobolev spaces] // Visn. Lviv. University.  Ser. mech-math.  [Herald of Lviv.University.  Ser. mech-math.].- Is. 64. - 2005. - Pages 44-61. 7. Pukach P.Ya. Mishana zadacha dlya odnoho neliniynoho rivnyannya typu kolyvan balky v neobmezheniy oblasti [Mixed problem for some nonlinear equation of beam vibrations type in unbounded domain] // Nauk. Visn. Cherniveckoho Nac. University. Ser. Matematika[Herald of Chernivci Nath. University.  Ser. Mathematics] - Is. 314 - 315. - 2006. - Pages 159 - 170. 8. Andrews K.T., Shillor M., Wright S. On the dynamic vibrations of an elastic beam in frictional contact with a rigid obstacle // Journ. of Elasticity - 42. - 1996. - P. 1 - 30. 9. Clark H.R. Elastic membrane equation in bounded and unbounded domains // Electr. Journ. of Qualit. Theory of Diff. Equat. - 2002. - No 7. - P. 1-21. 10. Messaoudi Salim A. Blow-up of positive-initial-energy solutions of nonlinear viscoelastic hyperbolic equation//Journ. of Math. Anal. and Appl. - 2006. - 320.- P. 902-915. 11. Koddington E.А., Levinson N. Teoriya obyknovennykh differentsialnykh uravneniy [Theory of ordinary differential equations].- Мoscow: Izd. inostr. lit., 1958.- 474 p. 12. Demidovich B.P. Lektsii po matematicheskoy teorii ustoychivosti [Lectures on the mathematical theory of stability].- Moscow: Nauka, 1967.- 472 p. 13. Lions  J.L. Nekotorye metody resheniya nelineynykh krayevykh zadach  [Some methods for sol¬ving nonlinear boundary value problems.- Moscow: Editorial URSS, 2002.- 587 p. 14. Pukach P.Ya. Pro neisnuvannya hlobalnoho za chasovoyu zminnoyu rozvyazku mishanoi zadachi dlya odnoho neliniynoho rivnyannya pyatoho poryadku [On the nonexistence of global by time variable solution of the mixed problem for some nonlinear equation of fifth order]// Prykladni problemy mekhaniky ta matematyky [Applied problems of mechanics and mathematics]. –2009. – Issue 7.–Pages 7 –15. Dzh. M.T. Tompson. Neustojchivosti i katastrofy v nauke i texnike [Instabilities and catastrophes in science and engineering].-Moscow: Mir., 1985.- 254 p.