Solutions to Inverse Problems in the field of Mechanical Vibration: Utilizing Pseudo-Spectral Method for the Dynamic Analysis of One-Dimensional Non-Ideal Cantilevers & Fixed Beams.

In this paper, we address the free vibration of non-ideal cantilevers and fixed beams in one dimension. According to the Euler-Bernoulli theory, a 4th order ODE models the free vibration of beams. The ODE is refined to an eigenvalue problem, where computing the natural frequencies become evident. The analytic approach for getting the natural frequencies is proven to be lengthy and problem-specific. Alternatively, we compute the natural frequencies of the beams using forward solvers that utilize the Pseudo-spectral method based on the Chebyshev polynomials of the first kind. The solvers are designed to output the natural frequencies of the beams subject to non-ideal boundary conditions (i.e supplying decay parameter(s) kL/kR as input(s)). Inverse solvers for the equivalent inverse problems are also designed, where the damage parameter(s) to be computed after supplying the natural frequencies of the beams as inputs. All solvers are written in MATLAB [3] language, and the computational approach is conclusively accurate. The codes can be modified to suit other beam configurations as well.