Fractal Dimensions in Elastic-Plastic Beam Dynamics
Abstract The final midpoint displacement of a two-degree-of-freedom beam model subjected to a short pulse of transverse loading may be either in the direction of the initial impulse or in the opposite (“negative”) direction, when moderately small plastic deformations occur. In the range where chaotic vibrations occur, the result depends with great sensitivity on the impulse magnitude. Considering a pulse of duration 0.5 × 10−3 sec, 100 calculations have been made for pulse forces P starting at 2500 N and increasing by increments of 2.0, 10−2, 10−4, and 10−6 N. It is found that the proportion and distribution of negative final displacements remain, on average, the same, independent of the size of the force increment. A fractal dimension representing a self-similarity property is calculated for the four choices of the force increment, and is found to be approximately 0.78 in each case. A correlation fractal dimension is also computed for undamped responses.