Engineering Transactions, 47, 1, pp. 21–37, 1999

Free Vibration of the System of Two Timoshenko Beams Coupled by a Viscoelastic Interlayer

K. Cabańska-Płaczkiewicz
Pedagogical University in Bydgoszcz
Poland

In this paper the uniform analytical method [3] has been used for solving a problem of free vibrations of continuous sandwich beam with damping. External layers are modelled as Timoshenko beams, while the internal layer possesses the characteristics of a viscoelastic, one-directional Winkler foundation. The phenomenon of free vibration has been described using a homogenous system of coupled partial differential equations. After separation of variables in the system of differential equations, the boundary problem has been solved and four complex sequences have been obtained: the sequences of frequencies, and the sequences of free vibration modes. Then, the property of orthogonality of complex free vibration modes has been demonstrated. The free vibration problem has been solved for arbitrarily assumed initial conditions.
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References

K. CABAŃSKA-PŁACZKIEWICZ, Dynamics of the system of two Bernoulli-Euler beams with a viscoelastic interlayer [in Polish], XXXVIIth Symp. of Modelling in Mech., 7, Silesian Univ. of Tech., 49-54, Gliwice 1998.

K. CABAŃSKA-PŁACZKIEWICZ, Free vibration of the string-beam system with a viscoelastic interlayer [in Polish], Theor. Found, in Civil Engng., 6, Faculty of Civil Engng., Warsaw Univ. of Tech., 59-68, Warsaw 1998.

K. CABAŃSKA-PŁACZKIEWICZ, Free vibration of the system of two strings coupled by a viscoelastic interlayer, J. Engng. Trans., 46, 2, 217-227, Warsaw 1998.

K. CABAŃSKA-PŁACZKIEWICZ, Free vibration of the system of two viscoelastic beams coupled by a viscoelastic interlayer, J. AkustiCnij Visnik, 1, 4, NANU, Institut Gidromechaniki, Kiev 1998 (in printing).

K. CABAŃSKA-PŁACZKIEWICZ, N. PANKRATOVA, The dynamic analysis of the system of two beams coupled by an elastic interlayer, XXXVIIIth Symp. of Modelling in Mech., 9, Silesian Univ. of Tech., 23-28, Gliwice 1999.

R. GUTOWSKI, Ordinary differential equations [in Polish], WNT, Warsaw 1971.

G. JEMIELITA, On the shear coefficients [in Polish], Theor. Found, in Civil Engng., 6, Faculty of Civil Engng., Warsaw Univ. of Tech., 115-122, Warsaw 1998.

G. JEMIELITA, Deformation of shear stiffness for a moderate thickness plate. Shear coefficients [in Polish], Theor. Found, in Civil Engng, 6, Faculty of Civil Engng., Warsaw Univ. of Tech., 123-128, Warsaw 1998.

S. KASPRZYK, Dynamics of a continuous system [in Polish], Pub. of AGH, Krakow 1989.

S. KUKLA, Free vibration of the system of two beams connected by many translational springs, J. of Sound and Vibr., 172, 1, 130-135, 1994.

W. NADOLSKI, A. PIELORZ, Shear waves in buildings subject to seismic loadings, Building and Environment, 16, 4, 279-285, 1980.

J. NIZIOŁ, J. SNAMINA, Free vibration of the discrete-continue us system with damping, J. of Theor. and Apl. Mech., 28, 1-2, 149-160, Warsaw 1990.

W. NOWACKI, Structural dynamics [in Polish], Arkady, Warsaw 1972.

Z. ONISZCZUK, Vibration analysis of the compound continuous systems with elastic constraints [in Polish], Pub. of the Rzesz6w Univ. of Tech., Rzeszów 1997.

Z. OSIŃSKI, Theory of vibration [in Polish], PWN, Warsaw 1978.

Z. OSIŃSKI, Damping of mechanical vibration [in Polish], PWN, Warsaw 1979.

A. PIELORZ, Discrete-continuous models in the analysis of low structures subject to kinematic excitations caused by transversal waves, J. of Theor. and Appl. Mech., 34, 3, 547-566, Warsaw 1996.

N. PANKRATOVA, B. NIKOLABV, E. ŚWITOŃSKI, Non-axisymmetrical deformation of flexible rotational shells in classical and improved statements, J. Engng. Mechanics, 3, 2, 89-96, Brno 1996.

N. PANKRATOVA, J. BERNAZKA, E. PANCHENKO, Improved models of stress-strained state calculations of multilayer noncircular cylindrical shells, XXXVTIth Symp. of Modelling in Mech., 6, Silesian Univ. of Tech., 281-286, Gliwice 1998.

W. SZCZEŚNIAK, Initial conditions in the dynamic problem of Timoshenko beam [in Polish], Building Engineering, 108, Pub. of the Warsaw Univ. of Tech., 99-143, Warsaw 1989.

W. SZCZEŚNIAK, The selection of problems of beams and shells subjected to inertial moving load [in Polish], Building Engineering, 125, Pub. of the Warsaw Univ. of Tech., Warsaw 1994.

W. SZCZEŚNIAK, The selection of railway problems [in Polish], Building Engineering, 129, Pub. of the Warsaw Univ. of Tech., Warsaw 1995.

W. SZCZEŚNIAK, Vibration of elastic sandwich and elastically connected double-beam system under moving loads [in Polish], Building Engineering, 132, Pub. of the Warsaw Univ. of Tech., 111-151, Warsaw 1998.

S.P. TIMOSHENKO, Vibration problems in engineering, D. Van Nostrad Comp., Pricenton 1956.

S.P. TIMOSHENKO, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, J. Phil. Mag. 6, 41, 744-746, 1921.

S.P. TIMOSHENKO, Discussion of the report on stress in railway track, J. Transactions of the American Society of Civil Engineers, 86, 537-541, 1923.

S.P.TIMOSHENKO, Method of analysis of statical and dynamical stresses in rail, Proceedings of the Second International Congress for Applied Mechanics, Zurich, Switzerland, 407-418, 1926.

S.P. TIMOSHENKO, D.H. YOUNG, W. WEAVER, Vibration problems in engineering, 4th Ed., Wiley, New York 1974.

F. TSE, MORSE, R. HINKLE, Mechanical vibrations, Theory and Applications, Allyn and Bacon, Boston 1978.

T.M. WANG, Natural frequencies of continuous Timoshenko beams, J. of Sound and Vibr. 13, 409-414, 1970.

E. WINKLER, Die Lehre von der Elastizitat und Festigkeit, Prag, Dominicus 1867.

M. WOŹNIAK, Railway embankment as the building foundation [in Polish], Mathematical Modelling, Scientific Treatises and Monographs, SGGW-AR, Warsaw 1991.

C. WOŹNIAK, Foundations of dynamics of deformable bodies [in Polish], PWN, Warsaw 1969.




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