Vibrations of Beams in the Elasto-Plastic and Geometrically Nonlinear Regime
This chapter presents models for beams vibrating with moderately large displacements and with elasto-plasticity. The beams are initially straight, homogeneous and isotropic, and oscillate always in one plane. A method to solve the equations of motion in t
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Introduction Chapter Objectives
Often the design of a structural or machine element requires an analysis in order to predict stresses and strains. A major goal of this analysis is to avoid failure in operation. Due to uncertainties - as not anticipated variations in the material properties, slight changes in the geometry, or unexpected loads - designers do not know exactly what stresses and strains the element will endure in practice. In addition, the physical/mathematical models used to represent reality are idealised approximations. The common way that engineers have to account for these uncertainties is to use a factor of safety in design. There are usually many feasible designs, where the element performs its function without failure, but it is desirable to choose a design that approaches objectives such as minimizing cost, minimizing weight or maximizing the natural frequency without increasing weight. The realization of such a goal generally contradicts a large safety factor. In order to achieve
D. J. Wagg et al. (eds.), Exploiting Nonlinear Behavior in Structural Dynamics © CISM, Udine 2012
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P. Ribeiro
a close to best design, whilst maintaining safety, it is important to reduce uncertainties and therefore it is desirable that the physical/mathematical model of the structural/machine element is close to reality. Vibrations with large amplitude, which may cause large strains and stresses, occur in a few systems due to large loads or to loads with a frequency component close to a resonance frequency. This is a particular important issue in thin structures. For example in aircraft there is an obvious desire for thin-walled structural components, which are likelier to experience vibrations with amplitude of the order of their thickness (Amabili (2008), Sathyamoorthy (1987), Smith et al. (1961), Teh (1982), White (1978)). Moreover, systems that in current engineering practice are not designed to experience nonlinear vibrations, may, if modelled more accurately, be designed more efficiently and still in such a way that they perform their function safely, taking advantage of nonlinear dynamic analysis. In spite of the popularity of linear models in engineering, the study of oscillations with large amplitudes requires geometrically nonlinear models (there are numerous publications on this issue that corroborate the former sentence, including Amabili (2008), Bennouna (1982), Kadiri et al. (2002), Han (1993), Mei (1973, 1976), Murphy et al. (1996), Ribeiro (2001, 2004b), Sarma and Varadan (1984a), Sathyamoorthy (1987), Touz´e et al. (2004), Wolfe (1995), Zavodney and Nayfeh (1989) and other works quoted in the remainder of this section). The solution of the nonlinear systems of equations of motion is quite often achieved by iterative methods, with repeated update of the nonlinear model. Additional difficulties of nonlinear analyses result from the facts that the superposition principle is not applicable and multiple solutions for the same parameter (e.g., same excitation frequency) can exist. Even when geomet
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