Transverse vibration of nonuniform Euler-Bernoulli beams on bounded Time scales

Abstract

In this article, we consider Euler-Bernoulli equation of transverse vibrations of nonuniform beams on bounded time scales $\mathbb{T}.$ We will give a description of all maximal dissipative, maximal accretive, self adjoint and other extensions of such operators.

Keywords

• Dissipative extensions,Self adjoint extensions,A boundary value space,Boundary condition,Nonuniform beam,Time scale

References

1. [1] Hilger S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Results in Math., (1990); 1818-1856:
2. [2] Agarwal R. P., Bohner M., and Li W.-T., Nonoscillation and Oscillation Theory for Functional Differential Equations, vol. 267 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2004.
3. [3] Jones M. A., Song B. and Thomas D. M., “Controlling wound healing through debridement,” Mathematical and Computer Modelling, vol. 40, no. 9-10, pp. 1057–1064, 2004.
4. [4] Spedding V., “Taming nature’s numbers,” New Scientist, vol. 179, no. 2404, pp. 28–31, 2003.
5. [5] Thomas D. M., Vandemuelebroeke L. and Yamaguchi K., “A mathematical evolution model for phytoremediation of metals,” Discrete and Continuous Dynamical Systems. Series B, vol. 5, no. 2, pp. 411–422, 2005.ker, Florida, 2004.
6. [6] Anderson D. R. , Guseinov Gusein Sh., and Hoffacker J. , Higher-order self-adjoint boundary-value problems on time scales, J. Comput. Appl. Math., 194 ;2; (2006) 309-342.
7. [7] Atici Merdivenci F. and Guseinov Gusein Sh., On Green’s functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math., 141;1-2; (2002) 75-99 .
8. [8] Bohner M. and Peterson A., Dynamic Equations on Time Scales, Birkhauser, Boston, 2001.

9. [9] Bohner M. and Peterson A., (Eds.), Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston, 2003.
10. [10] Gorbachuk M. L. and Gorbachuk V.I. , Boundary Value Problems for Operator Differential Equations, Naukova Dumka, Kiev, 1984; English transl. 1991;Birkhauser Verlag.
11. [11] Guseinov Gusein Sh., Self-adjoint boundary value problems on time scales and symmetric Green’s functions, Turkish J. Math., 29 (4); (2005) 365-380.
12. [12] Kochubei A. N., Extensions of symmetric operators and symmetric binary relations, Mat. Zametki 17; (1975) 41􀀀48; English transl. in Math. Notes 17; (1975) 25-28.
13. [13] M.G. Krein, On the indeterminate case of the Sturm-Liouville boundaryvalue problem in the interval (0;¥)’, Akad. Nauk SSSR Ser. Mat. 16; (1952) ; 292-324:
14. [14] F.G. Maksudov and B.P. Allahverdiev, On the extensions of Schr¨odinger operators with a matrix potentials, Dokl. Akad. Nauk 332; no.1; (1993) ;18- 20;English transl. Russian Acad. Sci. Dokl. Math. 48 (1994) ; no.2; 240-243:
15. [15] M. M. Malamud and V. I. Mogilevskiy, On extensions of dual pairs of operators, Dopov. Nats Akad. Nauk. Ukr. (1997) ;no. 1;30-37:
16. [16] B.P. Allahverdiev, Extensions of symmetric singular second-order dynamic operators on time scales. Filomat 30 (2016), no. 6, 1475–1484.
17. [17] B.P. Allahverdiev, Extensions of symmetric infinite Jacobi operator. Linear Multilinear Algebra 62 (2014), no. 9, 1146–1152.
18. [18] B.P. Allahverdiev, Extensions of symmetric second-order difference operators with matrix coefficients. J. Difference Equ. Appl. 19 (2013), no. 5, 839–849.
19. [19] A. Canoğlu and B.P. Allahverdiev, Selfadjoint and dissipative extensions of a symmetric Schr¨odinger operator. Math. Balkanica (N.S.) 17 (2003), no. 1-2, 113–120.
20. [20] H. Tuna and B. P. Allahverdiev, Dissipative Extensions of Fourth Order Differential Operators, Thai Journal of Mathematics, Vol. 16 (1), (2018), 275-285.
21. [21] Lakshmikantham V., Sivasundaram S. and Kaymakcalan B., Dynamic Systems on Measure Chains, Kluwer Academic Publishers, Dordrecht, 1996.
22. [22] Naimark M. A., Linear Differential Operators, 2nd edn., 1968; Nauka, Moscow, English transl. of 1st. edn., 1; 2, 1969; New York.
23. [23] Rynne B. P., L2 spaces and boundary value problems on time-scales, J. Math. Anal. Appl. 328; (2007) 1217-1236.
24. [24] V.M. Bruk, On a class of boundary –value problemswith a spectral parameter in the boundary conditions, Mat. Sb.,100; (1976) ;210-216: [25] J. W. Calkin, Abstract boundary conditions, Trans. Amer. Math. Soc.,Vol 45;No. 3; (1939) ;369-442:
25. [26] M.L. Gorbachuk, V.I. Gorbachuk and A.N. Kochubei, 1989. The theory of extensions of symmetric operators and boundary-value problems for differential equations’, Ukrain. Mat. Zh. 41; (1989) ;1299􀀀1312; English transl. in Ukrainian Math. J. 41(1989);1117-1129:
26. [27] J. von Neumann, Allgemeine Eigenwertheorie Hermitischer Functionaloperatoren, Math. Ann. 102; (1929) ;49-131.
27. [28] F.S. Rofe-Beketov, Self-adjoint extensions of differential operators in a space of vector valued functions’, Dokl. Akad. Nauk SSSR 184; (1969) ;1034-1037;English transl. in Soviet Math. Dokl. 10(1969);188-192
28. [29] H.D. Conway, J.F. Dubil, Vibration frequencies of truncated wedge and cone beam, Journal of Applied Mechanics 32E (1965) 932–935.
29. [30] J.J. Mabie, C.B. Rogers, Traverse vibrations of tapered cantilever beams with end support, Journal of Acoustical Society of America 44 (1968) 1739–1741.
30. [31] M.A. De Rosa, N.M. Auciello, Free vibrations of tapered beams with flexible ends, Computers & Structures 60 (2) (1996) 197–202.
31. [32] H.D. Conway, E.C.H. Becker, J.F. Dubil, Vibration frequencies of tapered bars and circular plates, Journal of Applied Mechanics June (1964) 329–331.
32. [33] E.T. Cranch, A. Adler, Bending vibrations of variable section beams, Journal of Applied Mechanics March (1956) 103–108.
33. [34] N.M. Auciello, G. Nole, Vibrations of a cantilever tapered beam with varying section properties and carrying a mass at the free end, Journal of Sound and Vibration, 214 (1) (1998) 105–119.
34. [35] H.C. Wang, Generalized hypergeometric function solutions on the transverse vibrations of a class of non-uniform beams, Journal of Applied Mechanics 34E (1967) 702–708.
35. [36] D. Storti, Y. Aboelnaga, Bending vibrations of a class of rotating beams with hypergeometric solutions, Journal of Applied Mechanics, 54 (1987) 311–314.
36. [37] D.I. Caruntu, On nonlinear vibration of nonuniform beam with rectangular cross-section and parabolic thickness variation, Solid Mechanics and its Applications, Vol. 73, Kluwer Academic Publishers, Dordrecht, Boston, London, 2000, pp. 109–118.
37. [38] D.I. Caruntu, Relied studies on factorization of the differential operator in the case of bending vibration of a class of beams with variable crosssection, Revue Roumaine des Sciences Techniques, S´erie de Mecanique Appliquee, 41 (5–6) (1996) 389–397.
38. [39] D.I. Caruntu, Dynamic modal characteristics of transverse vibrations of cantilevers of parabolic thickness. Mechanics Research Communications 36: (2009)391–404
39. [40] S. Naguleswaran, The vibration of a “complete” Euler–Bernoulli beam of constant depth and breadth proportional to axial co-ordinate raised to a positive exponents, Journal of Sound and Vibration, 187 (2) (1995) 311–327.
40. [41] S. Naguleswaran, A direct solution for the transverse vibration of Euler–Bernoulli wedge and cone beams, Journal of Sound and Vibration, 172 (3) (1994) 289–304.
41. [42] A.D. Wright, C.E. Smith, R.W. Thresher, J.L.C. Wang, Vibration modes of centrifugally stiffened beam, Journal of Applied Mechanics, 49 (1982) 197–202.
42. [43] Q. Wang, Sturm–Liouville equation for free vibration of a tube-in-tube tall building, Journal of Sound and Vibration, 191 (3) (1996), 349–355.