Dynamic stiffness formulation and free vibration analysis of a three-layered sandwich beam

A dynamic stiffness theory of a three-layered sandwich beam is developed and subsequently used to investigate its free vibration characteristics. This is based on an imposed displacement field so that the top and bottom layers behave like Rayleigh beams, whilst the central layer behaves like a Timoshenko beam. Using Hamilton's principle the governing differential equations of motion of the sandwich beam are derived for the general case when the properties of each layer are dissimilar. For harmonic oscillation the solutions of these equations are found in exact analytical form, taking full advantage of the application of symbolic computation, which has also been used to obtain the amplitudes of axial force, shear force and bending moment in explicit analytical forms. The boundary conditions for responses and loads at both ends of the freely vibrating sandwich beam are then imposed to formulate the dynamic stiffness matrix, which relates harmonically varying loads to harmonically varying responses at the ends. Using the Wittrick-Williams algorithm the natural frequencies and mode shapes of some representative problems are obtained and discussed. The important degenerate case of a symmetric sandwich beam is also investigated.