Abstract
Several structures of important industrial relevance feature large-amplitude vibration, which can have an initially hardening or a softening behavior. Models with quadratic and cubic nonlinear stiffness terms are often used to predict the nonlinear dynamics of such structures. If boundary conditions constitute an important cause of nonlinearity, however, it may not be possible to describe the nonlinear behavior of these systems by means of these nonlinear terms. The bundles of fuel rods immersed in the coolant flow of pressurized water reactors constitute an important example of extreme softening behavior, which is not described accurately by the simple adoption of nonlinear stiffness terms proportional to the square and to the cube of displacement in the equation of motion. In fact, the spacer grids, which support the nuclear fuel rods inside the reactors, constitute nonlinear boundary conditions, and are responsible of the peculiar softening behavior. Dedicated experiments on the rotational constraint applied by the spacer grids were performed, revealing a softening stiffness, which conforms extremely well with a bilinear model with hysteresis, such as the one formulated by Caughey. Therefore, a single degree-of-freedom equation with bilinear stiffness terms was employed to model the large-amplitude vibrations of single and bundled fuel rods immersed in quiescent water, which were detailed by previous stepped-sine experiments. A viscous dissipative term was initially retained for modelling damping, since several sources of dissipation are present for fuel rods immersed in water, besides hysteresis at the spacer grid supports. A satisfying fit of the experimental frequency- and time-domain curves was achieved through the use of an optimization algorithm for the tuning of stiffness and damping terms. Viscous damping, however, is not constant during large-amplitude vibrations, but is amplitude-dependent. Therefore, the value of the relevant parameter had to be changed at each forcing level. The model was then refined by introducing four additional switch points for stiffness, besides the initial one. Moreover, a nonlinear damping term of the quadratic type, suitable for softening structures immersed in fluids, was included. The resulting equation of motion with a five-switch piecewise linear stiffness describes with improved accuracy the experimental results. Also, one set of damping parameters is capable of describing the experimental results in the entire forcing (vibration amplitude) range. All this clarifies that a simple multilinear stiffness model, with the inclusion of quadratic damping, can be successfully employed to describe the behavior of rods immersed in fluid, in presence of nonlinear boundary conditions introduced by spacer grids.
