Keywords: sandwich beam, free vibration, dynamic stiffness.

Sandwich beams have widespread applications in engineering as load carrying structural members with high strength to weight ratios. This has stimulated continuing research in this area. In particular, the free vibration analysis of sandwich beams has been carried out by a number of investigators[1,2,3,4,5,6]. Di Taranto[1] and Mead and Marcus[2] are some of the earlier investigators who studied the free vibration problem of sandwich beams using classical approach. In essence they solved the governing differential equations of motion of a sandwich beam and imposed boundary conditions to obtain the natural frequencies and mode shapes. In later years, with the advent of digital computers, finite element based solutions became available[3,5]. One of the interesting features of the published literature in this area is that the value judgement made by various investigators when establishing the model accuracy of a composite beam has been interestingly, and often intriguingly, diverse. There is no doubt that there is considerable difficulty in obtaining an accurate analytical (mathematical) model for a sandwich beam. The difficulty arises from the very nature of the problem in which two or more structural components with different properties are joined together. For instance in the case of a three-layered sandwich beam consisting of a (soft) core and two face materials, the difficulty in the formulation would appear when preserving equilibrium and compatibility conditions in the interfaces between the core and the face materials. In the early eighties Mead[4] made a note worthy contribution in which he made an objective assessment of various sandwich beam models which were used to investigate the free vibration characteristics. Basically he compared the governing differential equations derived by various authors for free vibration analysis of sandwich beams. A relatively simple (but practically realistic) model assumes that the top and the bottom faces of a sandwich beam deform according to the Bernoulli- Euler beam theory whereas the core deforms only in shear. This model has been used by many[1,2] and has provided an important basis for further development on the subject. Apparently, the dynamic stiffness method, which has all the essential features of the finite element method, but has much better model accuracy, has not been applied to sandwich beams.

Thus the main purpose of this paper is to develop the dynamic stiffness matrix of a three-layered symmetric sandwich beam and then to use it to investigate its free vibration characteristics. The relatively simple model described above is taken in this preliminary, but novel study and the work has been greatly assisted by the symbolic computation package REDUCE[7]. The ensuing dynamic stiffness matrix is used in conjunction with the well-known Wittrick-Williams algorithm[8] to obtain the natural frequencies and mode shapes of an illustrative example taken from the literature. The results are discussed and some conclusions are drawn. It is expected that the present research will pave the way for further research on dynamic stiffness formulation of complex sandwich structural elements.

1

R.A DiTaranto, "Theory of Vibratory Bending for Elastic and Viscoelastic Layered Finite Length Beams", Journal of Applied Mechanics, 87, 881-886, 1965.

2

D.J Mead, S. Markus, "The Forced Vibration of a Three-Layer, Damped Sandwich Beam with Arbitrary Boundary Conditions", Journal of Sound and Vibration, 10(2), 163-175, 1969. doi:10.1016/0022-460X(69)90193-X

3

K.H. Ahmed, "Free Vibration of Curved Sandwich Beams by the method of finite elements", Journal of Sound and Vibration, 18(1), 61-74, 1971. doi:10.1016/0022-460X(71)90631-6

4

D.J. Mead, "A Comparison of some Equations for the Flexural Vibration of Damped Sandwich Beams", Journal of Sound and Vibration, 83(3), 363-377, 1982. doi:10.1016/S0022-460X(82)80099-0

5

T.T. Baber, R.A. Maddox, C.E. Orozco, "A Finite Element Model for Harmonically Excited Visoelastic Sandwich Beams", Computers & Structures, 66(1), 105-113, 1998. doi:10.1016/S0045-7949(97)00046-1

6

E.M. Austin, D.J. Inman, "Some Pitfalls of Simplified Modeling for Viscoelastic Sandwich Beams", Journal of Vibration and Acoustics, 122, 434-439, 2000. doi:10.1115/1.1287030

7

J. Fitch, "Solving algebraic problems with REDUCE", Journal of Symbolic Computing, 1, 211-227, 1985. doi:10.1016/S0747-7171(85)80015-8

8

W.H. Wittrick, F.W. Williams, "A General Algorithm for Computing Natural Frequencies of Elastic Structures", Quarterly Journal of Mechanics and Applied Mathematics, 24, 263-284, 1971.

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