Issue 10

A. Carpinteri et alii, Frattura ed Integrità Strutturale, 10 (2009) 3-11; DOI: 10.3221/IGF-ESIS.10.01 9 Several models have been proposed in the past for dealing with cracked vibrating beams [42-44], but, in all these models, the main assumption has been that the crack can be either fully open or fully closed during the vibration. Carpinteri and Pugno [10] recently developed a coupled theoretical and numerical approach to evaluate the nonlinear complex oscillatory behaviour in damaged structures under excitation. In their approach, they have focused their attention on a cantilever beam with several breathing transverse cracks and subjected to harmonic excitation perpendicular to its axis. The method, that is an extension of the super-harmonic analysis carried out by Pugno et al. [45] to subharmonic and zero frequency components, has allowed to capture the complex behavior of the nonlinear system, e.g., the occurrence of period doubling, as experimentally observed by Brandon and Sudraud [46] in cracked beams. A pioneer work on period doubling was written in 1978, when Mitchell Feigenbaum [47] developed a theory to treat the route from ordered to chaotic States. Even if oscillators showing the period doubling can be of different nature, as in mechanical, electrical, or chemical systems, they all share the characteristic of recursiveness. He provided a relationship in which the details of the recursiveness become irrelevant, through a kind of universal parameter, measuring the ratio of the distances between successive period doublings, the so called Feigenbaum's delta. His understanding of the phenomenon was later experimentally confirmed [48] , so that today we refer to the so-called Feigenbaum's period doubling cascade. However, even if the period doubling has a long history, only recently it has been experimentally observed in the dynamics of cracked structures [46]. To highlight the influence of the crack on the beam dynamics, let us consider two different numernical examples: a wikely nonlinear structure and a strongly nonlinear one. Only in the latter case the so called period doubling phenomenon clearly appears. Details about the beam geometry and materials can be found in [10]. For each of the two considered structures (Figs. 6a and 6b) the trajectory in the phase space is represented in Figs. 7a and 7b. In a hypothetical linear structure, the structural response is linear by definition with obviously only one harmonic component at the same frequency of the excitation. In the weakly nonlinear structure of Fig. 6a, the response converges and it appears only weakly nonlinear. The trajectory in the phase diagram is close to an ellipse. The diagram is nonsymmetric as the spatial positions of the cracks (placed in the upper part of the beam). The trajectory is an unique closed curve since here the period of the response is equal to the period of the excitation. Figure 6 : Damaged structures: weakly nonlinear (a) and strongly nonlinear (b). Figure 7 : Dimensionless phase diagram of the response (free end displacement): weakly (a) and strongly (b) non linear structure. In the strongly nonlinear structure of Fig. 6b t he nonlinearity increases. The harmonic components in the structural response are the zero one, the superharmonics as well as the subharmonic ones. It should be emphasized that a strong nonlinearity causes the period doubling of the response, i.e., the ω/2 component. The free-end vibrates practically with a period doubled with respect to the excitation. A nonnegligible component at ω/4 is observed too, representing a route to chaos through a period doubling cascade. The corresponding phase diagram clearly evidences this: the trajectory is composed by multiple cycles since here the period of the response is not equal to the period of the excitation. The