Rayleigh quotient and orthogonality in the linear space of boundary functions, finding accurate upper bounds of natural frequencies of non-uniform beams

Abstract

Instead of the traditional method of utilizing the extremal values of Rayleigh quotient, in this research we develop an upper bound theory to determine the natural frequencies in a linear space of boundary functions. The boundary function will have the following restrictions: boundary conditions are satisfied, fourth-order polynomial at least with unit leading coefficient. It proves that the maximum value of Rayleigh quotient in terms of kth-order boundary function builds a good upper bound when the $$(k - 3)$$th-order natural frequency is approximated. There are four boundary conditions of the beam, so the fourth-order boundary function is unique without having any parameter. On the contrary, the kth-order boundary function has $$k-4$$ free parameters, which leave us a chance to optimize them. We employ the orthogonality to derive the higher-order optimal boundary functions from the lower-order optimal boundary functions exactly, which are constructed sequentially by starting from the fourth-order boundary function. We examine the upper bound theory for uniform beams under three types of supports, and the single- and double- tapered beams under cantilevered support. Comparison is made between the first four natural frequencies with the exact/numerical ones, which proves the usefulness of the upper bound theory.