Recent Advances in the Numerical Integration of Multibody Systems

Andrzejewski, Thomas; Bock, Hans Georg; Eich, Edda; Schwerin, Reinhold von

doi:10.1007/978-94-017-0625-4_7

Cited by 18 publications

(8 citation statements)

References 9 publications

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“…(19), C Eq ξ = 0, is not considered in [3,39] which gives a sufficient but not a necessary condition that the mass matrix M be positive definite [34]. However, the positive definite mass matrix condition is too restrictive [34], and it is not always satisfied as shown in Eqs.…”

Section: Spatial Multibody Dynamics Equationsmentioning

confidence: 88%

“…This problem can be solved iteratively by the projection method based on the constrained and weighted least squares method as shown above, and two popular solutions [5,6,[17][18][19][20][21][23][24][25] will be given in the following subsections.…”

Section: Constrained and Weighted Least Squares Methodsmentioning

confidence: 99%

“…(19) gives that the accelerations and constraint forces are determined, but the Lagrange multipliers' vector λ E is not if matrix C Eq does not have full rank, which means that the coefficient matrix on the left of Eq. (15) is singular.…”

Section: Spatial Multibody Dynamics Equationsmentioning

confidence: 99%

“…A projection based on Lagrange multipliers was first proposed by Lubich [17] to modify the approximation position and velocity of a constrained multibody system after each time step, which is solved by the modified Newton iteration. Eich [18] proposed a constrained least-squares-based method which is similar to the method by Lubich when the mass matrix is used as a weight [5,6,19]. Yoon et al [20] developed a geometric elimination method of constraint violations using a gradient-based procedure where generalized coordinates were corrected by the geometric constraint equations and the generalized velocities were corrected by the energy constraint.…”

Section: Introductionmentioning

confidence: 99%

“…Blajer [23] developed a unified geometric formulation based on the geometric interpretation of constrained dynamics and pointed out that neglecting the inertial attributes may occasionally lead to physical inconsistency in mathematical formulation. The mass matrix used as a weight indicates an energy norm in the velocity level for the projection methods [5,6,17,19]. An improved scheme for one-step elimination of the energy constraint violation was proposed as well by Blajer [24].…”

Section: Introductionmentioning

confidence: 99%

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A constraint violation suppressing formulation for spatial multibody dynamics with singular mass matrix

Zhang

Liu

2015

Multibody Syst Dyn

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The constraint violation problem of spatial multibody systems is analyzed in this paper. The mass matrix is singular in the equations of motion when the Euler parameters with the normalization constraints are used to describe the orientation of the spatial rigid body. The constrained and weighted least-squares-based geometrical projection method is implemented to suppress the constraint violation during numerical integration, and the explicit correction formulation can be obtained by the block matrix inversion scheme. The mass matrix weighted correction formulation gives the physically consistent energy norm, but it needs the mass matrix to be positive definite. To extend the physically consistent correction formulation for solving spatial multibody systems' constraint violation problems with a singular mass matrix, a Modified Mass-Orthogonal Projection Method (MMOPM) and a Generalized Physical Orthogonal Projection Method (GPOPM) are proposed. MMOPM modifies the mass matrix directly by adding a penalty factor matrix which appears in the mass-orthogonal projection method and leads to a positive definite weight matrix that satisfies the block matrix inversion scheme condition. GPOPM is a generalization of the physical orthogonal projection method where the constrained least-squares method is weighted by the positive semi-definite mass matrix and the correction formulation is given by using the generalized block matrix inversion scheme. Numerical results show the feasibility and accuracy of the presented MMOPM and GPOPM. The constraints in position and velocity can reach machine precision during numerical integration. The elimination of violation of position constrains may require few iterations, while the violation of velocity constraints is removed in one step, and GPOPM is more accurate in velocity correction than MMOPM.

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