, HYBRID SYSTEM OF DIFFERENTIAL EQUATIONS DESCRIBING SYSTEMS OF SOLIDS ATTACHED TO A TIMOSHENKO BEAM // BSU bulletin. Mathematics, Informatics. - 2019. №1. . - С. 65-77.

HYBRID SYSTEM OF DIFFERENTIAL EQUATIONS DESCRIBING SYSTEMS OF SOLIDS ATTACHED TO A TIMOSHENKO BEAM

The article proposes a generalized mathematical model described by a hybrid sys- tem of differential equations of a prescribed structure for one class of mechanical systems consisting of a system of connected solids, elastically attached to a Ti- moshenko beam. Theoretical background of the study of free vibrations has been developed for a generalized mathematical model, in particular, an analytical and numerical method for constructing a frequency equation based on the consideration of a boundary value problem for the corresponding hybrid system of differential equations. In this case, natural frequencies are in fact eigenvalues for which there exists a solution to a boundary value problem. A calculated example is given that shows the reliability and versatility of the proposed method for studying free vibra- tions of mechanical systems, which are systems of connected solids elastically at- tached to a Timoshenko beam.

Timoshenko beam; boundary value problem; mathematical model; solid; hybrid system of differential equations.

Mizhidon A. D., Dabaeva M. Zh. (Tsytsyrenova M. Zh.) Obobshchennaya matematicheskaya model sistemy tverdykh tel, ustanovlennykh na uprugom sterzhne [A Generalized Mathematical Model of the System of Solids Mounted on an Elastic Rod]. Vestnik Vostochno-Sibirskogo gosudarstvennogo tekhnicheskogo universiteta. 2013. No. 6. Pp. 5–12.

Mizhidon A. D., Barguev S. G. Kraevaya zadacha dlya odnoi gibridnoi sistemy differentsialnykh uravnenii [Boundary Value Problem for One Hybrid System of Dif- ferential Equations]. Vestnik Buryatskogo gosudarstvennogo universiteta. 2013. V. 9. Matematika i informatika. Pp. 130–137.

Mizhidon A. D., Mizhidon K. A. Sobstvennye znacheniya dlya odnoi sistemy gibridnykh differentsialnykh uravnenii [Eigenvalues for One System of Hybrid Differ- ential Equations]. Siberian Electronic Mathematical Reports. 2016. V. 13. Pp. 911– 922.

Mizhidon A. D., Kharakhinov A. V. K issledovaniyu kraevoi zadachi dlya balki Timoshenko s uprugo prikreplennym tverdym telom [Towards the Study of a Boundary Value Problem for the Timoshenko Beam with an Elastically Attached Solid]. Vestnik Buryatskogo gosudarstvennogo universiteta. Matematika, informatika. 2016. No. 1. Pp. 88–101.

Mizhidon A. D., Kharakhinov A. V. Chastotnoe uravnenie dlya balki Timo- shenko s uprugo prikreplennym telom s dvumya stepenyami svobody [Frequency Equation for the Timoshenko beam with an Elastically Attached Body with Two De- grees of Freedom]. Vestnik Buryatskogo gosudarstvennogo universiteta. Matematika, informatika. 2016. No. 4. Pp. 61–68.

Vladimirov V. S. Obobshchennye funktsii v matematicheskoi fizike [Generalized Functions in Mathematical Physics]. Moscow: Nauka Publ., 1976. 280 p.

Kukla S. Application of Green Functions in Frequency Analysis of Timoshenko Beams with Oscillators. Journal of Sound and Vibration. 1997. V. 205. Iss. 3. Pp. 355– 363.

Majkut L. Free and Forced Vibration of Timoshenko Beams Described by Sin- gle Difference Equation. Journal of Theoretical and Applied Mechanics. 2009. V. 47. Iss. 1. Pp. 193–210.

Yesilce Y. Free and Forced Vibrations of an Axially-Loaded Timoshenko Mul- ti-Span Beam Carrying a Number of Various Concentrated Elements. Shock and Vibra- tion. 2012. No.19. Pp. 735–752.

Yesilce Y. Differential Transform Method and Numerical Assembly Technique for Free Vibration Analysis of the Axial-Loaded Timoshenko Multiple-Step Beam Car- rying a Number of Intermediate Lumped Masses and Rotary Inertias. Structural Engi- neering and Mechanics. 2015. V. 53. No. 3. Pp. 537–573.

Wu J. S., Chen D. W. Free Vibration Analysis of a Timoshenko Beam Carrying Multiple Spring–Mass Systems by Using the Numerical Assembly Technique. 2001. Vol. 50. Iss. 5. Pp. 1039–1058.

Magrab E. B. Natural Frequencies and Mode Shapes of Timoshenko Beams with Attachments. Journal of Vibration and Control. 2007. V. 13. Iss. 7. Pp. 905–934.

Xu S., Wang X. The Discrete Singular Convolution for Analyses of Elastic Wave Propagations in One-Dimensional Structures. Applied Mathematical Modeling. 2010. V. 34. Iss. 11. Pp. 3493–3508.

Mizhidon A. D., Barguev S. G. Kraevaya zadacha dlya odnoi gibridnoi sistemy differentsialnykh uravnenii [Boundary Value Problem for One Hybrid System of Dif- ferential Equations]. Vestnik Buryatskogo gosudarstvennogo universiteta. 2013. V. 9. Matematika i informatika. Pp. 130–137.

Mizhidon A. D., Mizhidon K. A. Sobstvennye znacheniya dlya odnoi sistemy gibridnykh differentsialnykh uravnenii [Eigenvalues for One System of Hybrid Differ- ential Equations]. Siberian Electronic Mathematical Reports. 2016. V. 13. Pp. 911– 922.

Mizhidon A. D., Kharakhinov A. V. K issledovaniyu kraevoi zadachi dlya balki Timoshenko s uprugo prikreplennym tverdym telom [Towards the Study of a Boundary Value Problem for the Timoshenko Beam with an Elastically Attached Solid]. Vestnik Buryatskogo gosudarstvennogo universiteta. Matematika, informatika. 2016. No. 1. Pp. 88–101.

Mizhidon A. D., Kharakhinov A. V. Chastotnoe uravnenie dlya balki Timo- shenko s uprugo prikreplennym telom s dvumya stepenyami svobody [Frequency Equation for the Timoshenko beam with an Elastically Attached Body with Two De- grees of Freedom]. Vestnik Buryatskogo gosudarstvennogo universiteta. Matematika, informatika. 2016. No. 4. Pp. 61–68.

Vladimirov V. S. Obobshchennye funktsii v matematicheskoi fizike [Generalized Functions in Mathematical Physics]. Moscow: Nauka Publ., 1976. 280 p.

Kukla S. Application of Green Functions in Frequency Analysis of Timoshenko Beams with Oscillators. Journal of Sound and Vibration. 1997. V. 205. Iss. 3. Pp. 355– 363.

Majkut L. Free and Forced Vibration of Timoshenko Beams Described by Sin- gle Difference Equation. Journal of Theoretical and Applied Mechanics. 2009. V. 47. Iss. 1. Pp. 193–210.

Yesilce Y. Free and Forced Vibrations of an Axially-Loaded Timoshenko Mul- ti-Span Beam Carrying a Number of Various Concentrated Elements. Shock and Vibra- tion. 2012. No.19. Pp. 735–752.

Yesilce Y. Differential Transform Method and Numerical Assembly Technique for Free Vibration Analysis of the Axial-Loaded Timoshenko Multiple-Step Beam Car- rying a Number of Intermediate Lumped Masses and Rotary Inertias. Structural Engi- neering and Mechanics. 2015. V. 53. No. 3. Pp. 537–573.

Wu J. S., Chen D. W. Free Vibration Analysis of a Timoshenko Beam Carrying Multiple Spring–Mass Systems by Using the Numerical Assembly Technique. 2001. Vol. 50. Iss. 5. Pp. 1039–1058.

Magrab E. B. Natural Frequencies and Mode Shapes of Timoshenko Beams with Attachments. Journal of Vibration and Control. 2007. V. 13. Iss. 7. Pp. 905–934.

Xu S., Wang X. The Discrete Singular Convolution for Analyses of Elastic Wave Propagations in One-Dimensional Structures. Applied Mathematical Modeling. 2010. V. 34. Iss. 11. Pp. 3493–3508.