Techniques for approximating a spatially varying Euler-Bernoulli model with a constant coefficient model

Developing accurate models to describe the behaviour of a physical system often results in differential equations with spatially varying coefficients. A notable example of this that appears in many applications is the Euler-Bernoulli beam equation for transverse vibrations. This equation with spatially varying coefficients, such as when the bending stiffness or mass per unit length varies along the length of the beam, is of interest in the current research. Methods for approximating the Euler-Bernoulli equation with periodically varying coefficients have been proposed yet there is still a need for methods that approximate the more general, non-periodically varying, cases. The goal of this research is to obtain a constant coefficient Euler-Bernoulli equation that accurately approximates the original spatially varying equation using an inverse problem approach. Obtaining such an approximation has advantages in control applications where a constant coefficient model is strongly preferred for computational efficiency. The motivation for this research stems from previous work by the authors on modelling cable-harnessed structures. The spatially varying equation is solved using the Lindstedt-Poincaré perturbation method and these results are used to determine the approximate model. Multiple inverse problem methods for determining the coefficients in the approximate model are considered including metric minimization, the modal participation factor (MPF), and the proper orthogonal decomposition (POD). Continuous version of POD and MPF methods are obtained. Several wrapping patterns and boundary conditions are considered for comparison and the results are in good agreement with analytical and finite element analysis (FEA) results.