According to a recent paper (Laulusa and Bauchau, 2008, “Review of Classical Approaches for Constraint Enforcement in Multibody Systems,” ASME J. Comput. Nonlinear Dyn., 3(1), 011004), Maggi’s formulation is a simple and stable way to solve the dynamic equations of constrained multibody systems. Among the difficulties of Maggi’s formulation, Laulusa and Bauchau quoted the need for an appropriate choice (and change, when necessary) of independent coordinates, as well as the high cost of computing and updating the basis of the tangent null space of constraint equations. In this paper, index-1 Lagrange’s equations are first considered, including the not-so-rare case of having a singular mass matrix and redundant constraints. The existence and uniqueness of solution for acceleration vector and Lagrange multipliers vector is studied in a very simple way. Then, following Von Schwerin (Von Schwerin, Multibody System Simulation. Numerical Methods, Algorithms and Software, Springer, New York, 1999), Maggi’s formulation is described as the most efficient way (in general) to solve these index-1 equations. Next, an improved double-step method, which implements the matrix transformations of Maggi’s formulation in an efficient way, is described. Finally, two large real-life examples are presented.

According to a recent paper by Laulusa and Bauchau [1], Maggi's formulation is a simple and stable way to solve the dynamic equations of constrained multibody systems. Among the difficulties of Maggi's formulation, Laulusa and Bauchau quoted the need for an appropriate choice (and change, when necessary) of independent coordinates, as well as the high cost of computing and updating the basis of the tangent null space of constraint equations.In this paper, index-1 Lagrange's equations are first considered, including the not-so-rare case of having a singular mass matrix and redundant constraints. The existence and uniqueness of solution for acceleration vector and Lagrange multipliers vector is studied in a very simple way. Then, following Von Schwerin [2], Maggi's formulation is described as the most efficient way (globally speaking) to solve these index-1 equations. Next, an improved double-step method, which implements the matrix transformations of Maggi's formulation in an efficient way, is described. Finally, two large real-life examples are presented.

The gradient-based design optimization of mechanical systems requires robust and efficient sensitivity analysis tools. The adjoint method is regarded as the most efficient semi-analytical method to evaluate sensitivity derivatives for problems involving numerous design parameters and relatively few objective functions. This paper presents a discrete version of the adjoint method based on the generalized-alpha time integration scheme, which is applied to the dynamic simulation of flexible multibody systems. Rather than using an ad hoc backward integration solver, the proposed approach leads to a straightforward algebraic procedure that provides design sensitivities evaluated to machine accuracy. The approach is based on an intrinsic representation of motion that does not require a global parameterization of rotation. Design parameters associated with rigid bodies, kinematic joints, and beam sectional properties are considered. Rigid and flexible mechanical systems are investigated to validate the proposed approach and demonstrate its accuracy, efficiency, and robustness.

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A great variety of formulations exist for the numerical simulation of rigid-body systems, particularly of medium-large systems such as vehicles. Topological formulations, which are considered to be the most efficient ones, are often cumbersome and not necessarily easy to implement. As a consequence, there is a lack of comparative evidence to support the performance of these formulations. In this paper, we present and compare three state-of-the-art topological formulations for multibody dynamics: generalized semirecursive, double-step semirecursive, and subsystem synthesis methods. We analyze the background, underlying principles, numerical efficiency, and accuracy of these formulations in a systematic way. A 28-degree-of-freedom, open-loop rover model and a 16-degree-of-freedom, closed-loop sedan car model are selected as study cases. Insight on the key aspects toward performance is provided.

Ten years ago, an original semi-recursive formulation for the dynamic simulation of large-scale multibody systems was presented by García de Jalón et al. (Advances in Computational Multibody Systems, pp. 1-23, 2005). By taking advantage of the cut-joint and rod-removal techniques through a double-step velocity transformation, this formulation proved to be remarkably efficient. The rod-removal technique was employed, primarily, to reduce the number of differential and constraint equations. As a result, inertia and external forces were applied to neighboring bodies. Those inertia forces depended on unknown accelerations, a fact that contributed to the complexity of the system inertia matrix. In search of performance improvement, this paper presents an approximation of rod-related inertia forces by using accelerations from previous time-steps. Additionally, a mass matrix partition is carried out to preserve the accuracy of the original formulation. Three extrapolation methods, namely, point, linear Lagrange and quadratic Lagrange extrapolation methods, are introduced to evaluate the unknown rod-related inertia forces. In order to assess the computational efficiency and solution accuracy of the presented approach, a general-purpose MATLAB/C/C++ simulation code is implemented. A 15-DOF, 12-rod sedan vehicle model with MacPherson strut and multi-link suspension systems is modeled, simulated and analyzed.

This article deals with the dynamic response optimization of mechanical systems, based on the computation of independent state sensitivities. Specifically, the dynamic behavior of a coach is analyzed in detail so as to improve its response in terms of handling and ride comfort behaviors. To that end, the coach is modeled as an 18DOF multibody system, whose equations of motion are posed using an efficient dynamic formulation based on Maggi's equations. Next, a direct-automatic differentiation approach for the computation of independent state sensitivities is applied. This allows one to quantify the effect of 19 design parameters on the vehicle dynamic response and to compute the design sensitivities or objective function gradients. Finally, handling and ride comfort objective functions are defined and are used to carry out a multi-objective suspension design optimization process, improving the vehicle response by 70% in an effective yet automatic way.

Algorithms for the sensitivity analysis of multibody systems are quickly maturing as computational and software resources grow. Indeed, the area has made substantial progress since the first academic methods and examples were developed. Today, sensitivity analysis tools aimed at gradient-based design optimization are required to be as computationally efficient and scalable as possible. This paper presents extensive verification of one of the most popular sensitivity analysis techniques, namely the direct differentiation method (DDM). Usage of such method is recommended when the number of design parameters relative to the number of outputs is small and when the time integration algorithm is sensitive to accumulation errors. Verification is hereby accomplished through two radically different computational techniques, namely manual differentiation and automatic differentiation, which are used to compute the necessary partial derivatives. Experiments are conducted on an 18-degree-of-freedom, 366-dependent-coordinate bus model with realistic geometry and tire contact forces, which constitutes an unusually large system within general-purpose sensitivity analysis of multibody systems. The results are in good agreement; the manual technique provides shorter runtimes, whereas the automatic differentiation technique is easier to implement. The presented results highlight the potential of manual and automatic differentiation approaches within general-purpose simulation packages, and the importance of formulation benchmarking.