On the use of the phase closure principle to calculate the natural frequencies of a rod or beam with nonlinear boundaries

The calculation of natural frequencies of rods and beams with stiffness boundaries is a common problem encountered by vibration engineers. For linear springs attached to the boundaries, this is a straightforward problem, and the natural frequencies can be calculated using several well-known approaches. One approach that facilitates insight into the way in which the boundaries affect the natural frequencies of the system, is the use of the phase-closure principle. In this paper, this approach is applied to a rod and a beam with nonlinear stiffness boundaries, where the nonlinearity is of the hardening or softening type. The phases of the reflection coefficients, which are a function of frequency and vibration amplitude for a nonlinear boundary, are first calculated. They are then used in the application of the phase closure principle to determine the natural frequencies of the system. To illustrate the approach, examples are presented for both the rod and the beam. It is shown that while the hardening nonlinearity has a marginal effect in both cases, the softening nonlinearity can have a profound effect on the phase of the reflection coefficient, inducing some instabilities.