Free vibration of sandwich beams using the dynamic stiffness method

The free vibration analysis of symmetric sandwich beams is carried out in this paper by using the dynamic stiffness method. First the governing partial differential equations of motion of a three-layered symmetric sandwich beam undergoing free natural vibration are derived using Hamilton's principle. The formulation led to two partial differential equations that are both coupled in axial and bending deformations. While seeking solution for harmonic oscillation, the two equations are combined into one sixth-order ordinary differential equation, which applies to both axial and bending displacements. This procedure was facilitated by the use of symbolic computation with the package REDUCE. Closed form analytical solution of the sixth order differential equation is then obtained in its most general form in terms of six arbitrary constants. Expressions for axial force, shear force and bending moment are also obtained in terms of the six arbitrary constants. Next the boundary conditions for displacements and forces at the ends of the sandwich beam are applied to eliminate the constants. This essentially casts the equations in the form of element dynamic stiffness matrix of the sandwich beam relating harmonically varying forces with harmonically varying displacements. The resulting dynamic stiffness matrix is then applied in conjunction with the Wittrick-Williams algorithm to compute the natural frequencies and mode shapes of an example sandwich beam. Numerical results are discussed and compared with those available in the literature. This is followed by conclusions.