Development of Characteristic Orthogonal Polynomials (COPs) in Vibration Problems
Vibration analysis of different shaped structures has been of interest to several engineering disciplines over several decades. Dynamic behavior of the structures is strongly dependent on the boundary conditions, geometrical shapes, material properties, different theories, and various complicating effects. The importance of orthogonal polynomials is well known in the problems of numerical approximation. The orthogonal nature of the polynomials makes the analysis simple and straightforward. Applying the condition of stationarity of the natural frequencies at the natural modes, the variation of natural frequencies with respect to the arbitrary constants is equated to zero to obtain an eigenvalue problem. The vibration of structures such as rectangular plates whose deflection can be assumed in the form of product of one-dimensional characteristic orthogonal polynomials (COPs) may be analyzed using one-dimensional COPs. Polynomials are generated by using again the Gram–Schmidt orthogonalization procedure in some variables. Various studies related to the vibration problems of plates, beams, and with other complicating effects are handled using COPs.