Efficient and accurate modeling of rigid rods

Abstract

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.

An Efficient High-Order Time-Step Algorithm With Proportional-Integral Control Strategy for Semirecursive Vehicle Dynamics

“…Previous work related to the constant time-step integration schemes are, in general, easy to implement and suitable for efficient dynamic simulation. [9]- [17]. Although the further improvement related to advanced multibody formulations would be helpful for efficient simulation, their implementation may be challenging [17].…”

“…[9]- [17]. Although the further improvement related to advanced multibody formulations would be helpful for efficient simulation, their implementation may be challenging [17]. In practice, improvement for computational efficiency can be obtained by employing efficient implementations and time integration algorithms, as for example, sparse matrix technique [11], iterative algorithm [12], [18] and adaptive time-step integration scheme.…”

“…Its configuration can be represented by a set of n dependent relative coordinates z [8]. The equations of motion for a system under investigation can be written in terms of a small set of f independent relative accelerationsz i as follows [17]:…”

“…where T ∈ R 6n×6n is the path matrix representing the open-loop multibody system tree topology [24]; matrices R d ∈ R 6n×n and R z ∈ R n×f contain the first and second velocity transformation matrices, respectively;M ∈ R 6n×6n is the inertia matrix; M ∈ R 6n×6n and Q ∈ R 6n contain the accumulated inertia matrix and applied force vector, respectively; M rods ∈ R 6n×6n and Q rods ∈ R 6n contain rod contributions (slender rods are eliminated to open the closed-chain system; thus their inertia should be taken into account later) and P rods contains the velocity dependent inertia forces resulting from the eliminated rods. Further detailed explanations are available in [17]. This formulation, which is called the double-step semirecursive formulation, was proposed by García de Jalón et al The technical originality of the double-step semirecursive approach is associated to the efficient implementation of recursive kinematics and dynamics, the robust enforcement of constraint equations, the ordinary differential nature and the smallest set of the resulting equations.…”

“…To study the solution accuracy and computational efficiency of the presented adaptive time-step integration scheme, this section introduces a 15-DOF sedan vehicle and a 17-DOF light truck models. In practice, the multibody system modeling is accomplished by implementing the double-step semirecursive method with a rod-removal technique [17]. To make full use of the recursive kinematics and dynamics, a number of auxiliary massless bodies and 1-DOF revolute and prismatic joints are introduced to describe universal, spherical and free joints.…”