This paper presents the consistency and stability analyses of the Generalized-a methods applied to nonlinear dynamical systems. The second-order accuracy of this class of algorithms is proved also in the non-linear regime, independently of the quadrature rule for nonlinear internal forces. Conversely, the G-stability notion which is suitable for linear multistep schemes devoted to non-linear dynamic problems cannot be applied, as the non-linear structural dynamics equations are not contractive. Nonetheless, it is proved that the Generalized-a methods are endowed with stability in an energy sense and guarantee energy decay in the high-frequency range as well as asymptotic annihilation. However, overshoot and heavy energy oscillations in the intermediate-frequency range are exhibited. The results of representative numerical simulations performed on relatively simple single-and multiple-degrees-of-freedom non-linear systems are presented in order to confirm the analytical estimates. IntroductionFor many problems in structural dynamics, the time integration of stiff ordinary differential equations is required (Hairer and Wanner 1991, p. 9). Commonly used methods for integrating equations with timescales that differ by several orders of magnitude are implicit as relatively large time steps can be employed. As a matter of fact, most integration schemes are A-stable, i.e. unconditionally stable in the linear regime. Moreover, it is essential that these methods be endowed with mechanisms entailing numerical dissipation in the high-frequency range, with limited algorithmic damping in the low-frequency range. These mechanisms help to eliminate high-frequency modes that are insufficiently resolved by either the spatial discretization, the selected time step or both. Representative members of these algorithms are, among others, the N-b method (Newmark 1959), the HHT-a method (Hilber et al. 1977), the WBZ-a method (Wood et al. 1981), the HP-h 1 method (Hoff and Pahl 1988a, b) and the CH-a method (Chung and Hulbert 1993). These methods exhibit second order accuracy in linear dynamics and permit efficient variable step size techniques, being one-step methods. The CH-a, the HHT-a and the WBZ-a methods, the so-called a-methods, are one-parameter schemes which can be considered as particular cases of a more general class of methods named Generalized-a (G-a) in the foregoing. This class of methods corresponds to the CH-a scheme (Chung and Hulbert 1993), where the algorithmic parameters a m ; a f ; b and c are assumed to be independent of each other.For stiff linear problems, A-stability may not be sufficient to ensure a robust temporal integration. As a matter of fact, some stiff components of the numerical solution damp out very slowly even in the presence of numerical dissipation and can show up oscillations which alter the solution. The low effectiveness of the numerical dissipation and the overshoot consequences on the response of the HHT-a method applied to stiff dynamical systems have been highlighted by Bauchau et al. (1995)...

Several spring-damper-mass models of the human body have been developed in order to reproduce the measured ground vertical reaction forces during human running (McMahon and Cheng, 1990; Ferris et al., 1999; Liu and Nigg, 2000). In particular, Liu and Nigg introduced at the lower level of their model, i.e. at the interface between the human body and the ground, a nonlinear element representing simultaneously the shoe midsoles and the ground flexibility. The ground reaction force is modelled as the force supported by this nonlinear element, whose parameters are identified from several sets of experimental data. This approach proved to be robust and quite accurate. However, it does not explicitly take into account the shoe and the ground properties. It turns out to be impossible to study the influence of shoe materials on the impact force, for instance for footwear design purposes. In this paper, a modification of the Liu and Nigg's model is suggested, where the original nonlinear element is replaced with a bi-layered spring-damper-mass model: the first layer represents the shoe midsole and the second layer is associated with the ground. Ground is modelled as an infinite elastic half-space. We have assumed a viscoelastic behaviour of the shoe material, so the damping of shoe material is taken into account. A methodology for the shoe-soles characterization is proposed and used together with the proposed model. A parametric study is then conducted and the influence of the shoe properties on the impact force is quantified. Moreover, it is shown that impact forces are strongly affected by the ground stiffness, which should therefore be considered as an essential parameter in the footwear design.

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