Section: New Results

Numerical analysis of multibody mechanical systems with constraints

This scientific theme concerns the numerical analysis of mechanical systems with bilateral and unilateral constraints, with or without friction [1]. They form a particular class of dynamical systems whose simulation requires the development of specific methods for analysis and dedicated simulators [6].

The PhD thesis of N. Akadkhar under contract with Schneider Electric has concerned the numerical simulation of mechanical systems with unilateral constraints and friction, where the presence of clearances in imperfect joints plays a crucial role. A first work deals with four-bar planar mechanisms with clearances at the joints, which induce unilateral constraints and impacts, rendering the dynamics nonsmooth. The objective is to determine sets of parameters (clearance value, restitution coefficients, friction coefficients) such that the system's trajectories stay in a neighborhood of the ideal mechanism (i.e. without clearance) trajectories. The analysis is based on numerical simulations obtained with the projected Moreau-Jean time-stepping scheme. Circuits breakers with 3D joint clearances have been studied in [3] [41] where it is demonstrated that the nonsmooth dynamics approach as coded in our software siconos , allows a very good prediction of the system's dynamics, with experimental validation. An overview of various approaches for the feedback control of multibody systems with joint clearances is proposed in [4].

This work [16] concerns a formalism for the transient simulation of nonsmooth dynamic mechanical systems composed of rigid and flexible bodies, kinematic joints and frictionless contact conditions. The proposed algorithm guarantees the exact satisfaction of the bilateral and unilateral constraints both at position and velocity levels. Thus, it significantly differs from penalty techniques since no penetration is allowed. The numerical scheme is obtained in two main steps. Firstly, a splitting method is used to isolate the contributions of impacts, which shall be integrated with only first-order accuracy, from smooth contributions which can be integrated using a higher order scheme. Secondly, following the idea of Gear, Gupta and Leimkuhler, the equation of motion is reformulated so that the bilateral and unilateral constraints appear both at position and velocity levels. After time discretization, the equation of motion involves two complementarity conditions and it can be solved at each time step using a monolithic semi-smooth Newton method. The numerical behaviour of the proposed method is studied and compared to other approaches for a number of numerical examples. Itis shown that the formulation offers a unified and valid approach for the description of contactconditions between rigid bodies as well as between flexible bodies.

Collisions in musical string instruments play a fundamental role in explaining the sound production in various instruments such as sitars, tanpuras and electric basses. Contacts occuring during the vibration provide a nonlinear effect which shapes a specific tone due to energy transfers and enriches the hearing experience. As such, they must be carefully simulated for the purpose of physically-based sound synthesis. Most of the numerical methods presented in the literature rely on a compliant modeling of the contact force between the string and the obstacle. In this contribution, numerical methods from nonsmooth contact dynamics are used to integrate the problem in time. A Moreau-Jean time-stepping scheme is combined with an exact scheme for phases with no contact, thus controlling the numerical dispersion. Results for a two-point bridge mimicking a tanpura and an electric bass are presented, showing the ability of the method to deal efficiently with such problems while invoking, as compared to a compliant approach, less modelling parameters and a reduced computational burden [7].

In [15] report, we review several formulations of the discrete frictional contact problem that arises in space and time discretized mechanical systems with unilateral contact and three-dimensional Coulomb’s friction. Most of these formulations are well–known concepts in the optimization community, or more generally, in the mathematical programming community. To cite a few, the discrete frictional contact problem can be formulated as variational inequalities, generalized or semi–smooth equations, second–order cone complementarity problems, or as optimization problems such as quadratic programming problems over second-order cones. Thanks to these multiple formulations, various numerical methods emerge naturally for solving the problem. We review the main numerical techniques that are well-known in the literature and we also propose new applications of methods such as the fixed point and extra-gradient methods with self-adaptive step rules for variational inequalities or the proximal point algorithm for generalized equations. All these numerical techniques are compared over a large set of test examples using performance profiles. One of the main conclusion is that there is no universal solver. Nevertheless, we are able to give some hints to choose a solver with respect to the main characteristics of the set of tests

In [18] several “classical” multiple-impact laws are compared on chains of aligned balls: Moreau's law, the binary collision law, and the LZB approach [2]. Short analyses of these laws are made, and thorough comparisons are led numerically. It is concluded that both Moreau and the binary collision laws, furnish good results (in terms of predictability) only in very particular cases of elasticity coefficien, contact stiffenesses ratios, and mass ratios.