Simple formulas for the natural frequencies of non-uniform cables and beams

Reference Value Selection in a Perturbation Theory Applied to Nonuniform Beams

“…Therefore, another contribution of the presented work is exploring three methods, one gradient minimization and two -norm minimization methods, by which the reference values in the perturbation theory can be chosen. One method for finding these reference values is considered in [21]. Further, the proposed gradient minimization method in the presented work extends the results in [21] in a more general form to allow for an approach for systems where more than two reference values are required.…”

“…The assumption of the mass per unit length and bending stiffness being discontinuous at a set of points extends the work of [21] and constitutes one of the contributions of the current work. Accounting for a system with piecewise continuous properties will allow for future studies of more complex systems, such as cable-harnessed structures motivated by the space applications [22,23].…”

“…While quite a powerful method, this approach primarily builds on the assumption of periodic variations in the system parameters over the length of the structure. The transverse vibration of continuous nonuniform Euler-Bernoulli beams with an axial force was studied in [20,21], and no additional assumptions on the system, such as periodicity, were imposed. The WKB approximation was employed in [20] and the small parameter multiplying the highest derivative was related to the inverse of the natural frequencies of the system.…”

“…Although this proves to be an advantage for higher frequencies, the method becomes less effective for smaller frequencies. The Lindstedt-Poincaré method was first applied to a nonuniform Euler-Bernoulli beam in [21]. However, the work in [21] was restricted to continuous systems due to lack of continuity conditions at points of discontinuity.…”

“…The Lindstedt-Poincaré method was first applied to a nonuniform Euler-Bernoulli beam in [21]. However, the work in [21] was restricted to continuous systems due to lack of continuity conditions at points of discontinuity. In the current work, the LindstedtPoincaré method is applied to a nonuniform Euler-Bernoulli beam with piecewise continuous properties.…”