A variant of the generalized-α scheme is proposed for constrained mechanical systems represented by index-3 DAEs. Based on the analogy with linear multistep methods, an elegant convergence analysis is developed for this algorithm. Second-order convergence is demonstrated both for the generalized coordinates and the Lagrange multipliers, and those theoretical results are illustrated by numerical tests.

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SUMMARYIn flexible multibody dynamics, advanced modelling methods lead to high-order non-linear differentialalgebraic equations (DAEs). The development of model reduction techniques is motivated by control design problems, for which compact ordinary differential equations (ODEs) in closed-form are desirable. In a linear framework, reduction techniques classically rely on a projection of the dynamics onto a linear subspace. In flexible multibody dynamics, we propose to project the dynamics onto a submanifold of the configuration space, which allows to eliminate the non-linear holonomic constraints and to preserve the Lagrangian structure. The construction of this submanifold follows from the definition of a global modal parameterization (GMP): the motion of the assembled mechanism is described in terms of rigid and flexible modes, which are configuration-dependent. The numerical reduction procedure is presented, and an approximation strategy is also implemented in order to build a closed-form expression of the reduced model in the configuration space. Numerical and experimental results illustrate the relevance of this approach.

This paper proposes a family of Lie group time integrators for the simulation of flexible multibody systems. The method provides an elegant solution to the rotation parametrization problem. As an extension of the classical generalized-α method for dynamic systems, it can deal with constrained equations of motion. Second-order accuracy is demonstrated in the unconstrained case. The performance is illustrated on several critical benchmarks of rigid body systems with high rotation speeds, and second-order accuracy is evidenced in all of them, even for constrained cases. The remarkable simplicity of the new algorithms opens some interesting perspectives for real-time applications, model-based control, and optimization of multibody systems.

An original signal processing algorithm is presented to automatically extract, on a stride-by-stride basis, four consecutive fundamental events of walking, heel strike (HS), toe strike (TS), heel-off (HO), and toe-off (TO), from wireless accelerometers applied to the right and left foot. First, the signals recorded from heel and toe three-axis accelerometers are segmented providing heel and toe flat phases. Then, the four gait events are defined from these flat phases. The accelerometer-based event identification was validated in seven healthy volunteers and a total of 247 trials against reference data provided by a force plate, a kinematic 3D analysis system, and video camera. HS, TS, HO, and TO were detected with a temporal accuracy ± precision of 1.3 ms ± 7.2 ms, -4.2 ms ± 10.9 ms, -3.7 ms ± 14.5 ms, and -1.8 ms ± 11.8 ms, respectively, with the associated 95% confidence intervals ranging from -6.3 ms to 2.2 ms. It is concluded that the developed accelerometer-based method can accurately and precisely detect HS, TS, HO, and TO, and could thus be used for the ambulatory monitoring of gait features computed from these events when measured concurrently in both feet.

SUMMARY Mechanical systems are usually subjected not only to bilateral constraints, but also to unilateral constraints. Inspired by the generalized‐ α time integration method for smooth flexible multibody dynamics, this paper presents a nonsmooth generalized‐ α method, which allows a consistent treatment of the nonsmooth phenomena induced by unilateral constraints and an accurate description of the structural vibrations during free motions. Both the algorithm and the implementation are illustrated in detail. Numerical example tests are given in the scope of both rigid and flexible body models, taking account for both linear and nonlinear systems and comprising both unilateral and bilateral constraints. The extended nonsmooth generalized‐ α method is verified through comparison to the traditional Moreau–Jean method and the fully implicit Newmark method. Results show that the nonsmooth generalized‐ α method benefits from the accuracy and stability property of the classical generalized‐ α method with controllable numerical damping. In particular, when it comes to the analysis of flexible systems, the nonsmooth generalized‐ α method shows much better accuracy property than the other two methods. Copyright © 2013 John Wiley & Sons, Ltd.

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