Coupled Bending-Longitudinal Vibration of Three Layer Sandwich Beam using Exact Dynamic Stiffness Matrix

Document Type: Research Paper

Authors

1 School of Engineering, Yasouj University, Yasouj, Iran

2 Sahand University of Technology, Tabriz, Iran

3 Independent Consultant, Gwanwyn, Craig Penlline, CF71 7RT, UK

Abstract

A Newtonian (vectorial) approach is used to develop the governing differential equations of motion for a three layer sandwich beam in which the uniform distribution of mass and stiffness is dealt with exactly. The model allows for each layer of material to be of unequal thickness and the effects of coupled bending and longitudinal motion are accounted for. This results in an eighth order ordinary differential equation whose closed form solution is developed into an exact dynamic member stiffness matrix (exact finite element) for the beam. Such beams can then be assembled to model a variety of structures in the usual manner. However, such a formulation necessitates the solution of a transcendental eigenvalue problem. This is accomplished using the Wittrick-Williams algorithm, whose implementation is discussed in detail. The algorithm enables any desired natural frequency to be converged upon to any required accuracy with the certain knowledge that none have been missed. The accuracy of the method is then confirmed by comparison with five sets of published results together with a further example that indicates its range of application.  A number of further issues are considered that arise from the difference between sandwich beams and uniform single material beams, including the accuracy of the characteristic equation, co-ordinate transformations, modal coupling and the application of boundary conditions. 

Keywords


[1] Kerwin E. M., 1959, Damping of flexural waves by a constrained viscoelastic layer, Journal of the Acoustical Society of America 31: 952-962.
[2] Mead D. J., 1962, The Double-Skin Damping Configuration, University of Southampton.
[3] DiTaranto R. A., 1965, Theory of vibratory bending for elastic and viscoelastic layered finite-length beams, Journal of Applied Mechanics 32: 881-886.
[4] Yin T. P., Kelly T. J., Barry J. E., 1967, A quantitative evaluation of constrained layer damping, Transactions of the American Society of Mechanical Engineers, Journal of Engineering for Industry 89: 773-784.
[5] Mead D. J., Markus S., 1969, The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions, Journal of Sound and Vibration 10: 163-175.
[6] Rao D. K., 1978, Frequency and loss factors of sandwich beams under various boundary conditions, Journal of Mechanical Engineering Science 20: 271-282.
[7] Sakiyama T., Matsuda H., Morita C., 1996, Free vibration analysis of sandwich beam with elastic or viscoelastic core by applying the discrete Green function, Journal of Sound and Vibration 191: 189-206.
[8] Yu Y.Y., 1959, A new theory of elastic sandwich plate-one dimensional case, Journal of Applied Mechanics 26: 415-421.
[9] Rao Y. V. K. S., Nakra B. C., 1970, Influence of rotary and longitudinal translatory inertia on the vibrations of unsymmetrical sandwich beams, Proceeding of the 15th Conference I.S.T.A.M.
[10] Mead D. J., 1982, A comparison of some equations for the flexural vibration of damped sandwich beams, Journal of Sound and Vibration 83: 363-377.
[11] Chonan S., 1982, Vibration and stability of sandwich beams with elastic bonding, Journal of Sound and Vibration 85(4): 525-537.
[12] Mead D. J., Markus S., 1985, Coupled flexural, longitudinal and shear wave motion in two- and three-layered damped beams, Journal of Sound and Vibration 99(4): 501-519.
[13] Marur S. R., Kant T., 1996, Free vibration analysis of fiber reinforced composite beams using higher order theories and finite element modeling, Journal of Sound and Vibration 194: 337-351.
[14] Kameswara Rao M., Desai Y.M., Chitnis M. R., 2001, Free vibration of laminated beams using mixed theory, Composite Structures 52: 149-160.
[15] Silverman I. K., 1995, Natural frequencies of sandwich beams including shear and rotary effects, Journal of Sound and Vibration 183: 547-561.
[16] Fasana A., Marchesiello S., 2001, Rayleigh-Ritz analysis of sandwich beams, Journal of Sound and Vibration 241: 643-652.
[17] Amirani M. C., Khalili S. M. R., Nemati N., 2009, Free vibration analysis of sandwich beam with FG core using the element free Galerkin method, Composite Structures 90: 373-379.
[18] Hashemi S. M., Adique E. J., 2009, Free vibration analysis of sandwich beams: A dynamic finite element, International Journal of Vehicle Structures and Systems 1(4): 59-65.
[19] Hashemi S. M., Adique E. J., 2010, A quasi-exact dynamic finite element for free vibration analysis of sandwich beams, Applied Composite Materials 17(2): 259-269.
[20] Banerjee J. R., 2003, Free vibration of sandwich beams using the dynamic stiffness method, Computers and Structures 81: 1915-1922.
[21] Howson W. P., Zare A., 2005, Exact dynamic stiffness matrix for flexural vibration of three-layered sandwich beams, Journal of Sound and Vibration 282: 753-767.
[22] Banerjee J. R., Sobey A.J., 2005, Dynamic stiffness formulation and free vibration analysis of a three-layered sandwich beam, International Journal of Solids and Structures 42(8): 2181-2197.
[23] Banerjee J. R., Cheung C. W., Morishima R., Perera M., Njuguna J., 2007, Free vibration of a three-layered sandwich beam using the dynamic stiffness method and experiment, International Journal of Solids and Structures 44: 7543-7563.
[24] Jun L., Xiaobin L., Hongxing H., 2009, Free vibration analysis of third-order shear deformable composite beams using dynamic stiffness method, Archive of Applied Mechanics 79: 1083-1098.
[25] Khalili S. M. R., Nemati N., Malekzadeh K., Damanpack A. R., 2010, Free vibration analysis of sandwich beams using improved dynamic stiffness method, Composite Structures 92: 387-394.
[26] Damanpack A. R., Khalili S. M. R., 2012, High-order free vibration analysis of sandwich beams with a flexible core using dynamic stiffness method, Composite Structures 94: 1503-1514.
[27] Wittrick W. H., Williams F. W., 1971, A general algorithm for computing natural frequencies of elastic structures, Quarterly Journal of Mechanics and Applied Mathematics 24: 263-284.
[28] Howson W. P., Williams F. W., 1973, Natural frequencies of frames with axially loaded Timoshenko members, Journal of Sound and Vibration 26: 503-515.
[29] Zare A., 2004, Exact Vibrational Analysis of Prismatic Plate and Sandwich Structures, Ph.D. Thesis, Cardiff University.
[30] Ahmed K. M., 1971, Free vibration of curved sandwich beams by the method of finite elements, Journal of Sound and Vibration 18: 61-74.
[31] Ahmed K. M., 1972, Dynamic analysis of sandwich beams, Journal of Sound and Vibration 10: 263-276.
[32] Sakiyama T., Matsuda H., Morita C., 1997, Free vibration analysis of sandwich arches with elastic or viscoelastic core and various kinds of axis-shape and boundary conditions, Journal of Sound and Vibration 203(3): 505-522.
[33] Bozhevolnaya E., Sun J. Q., 2004, Free vibration analysis of curved sandwich beams, Journal of Sandwich Structures & Materials 6(1): 47-73.
[34] Petrone F., Garesci F., Lacagnina M., Sinatra R., 1999, Dynamical joints influence of sandwich plates, Proceedings of the 3rd European Nonlinear Oscillations Conference, Copenhagen, Denmark.
[35] Marura S. R., Kant T., 2008, Free vibration of higher-order sandwich and composite arches, Part I: Formulation, Journal of Sound and Vibration 310: 91-109.