Energy considerations for the stabilization of constrained mechanical systems with velocity projection

Orden, Juan Carlos García

doi:10.1007/s11071-009-9579-8

Cited by 16 publications

(8 citation statements)

References 19 publications

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“…To verify the feasibility and accuracy of the developed constraint violation suppressing formulation, two benchmark problems are presented in this section, a five-bar pendulum system [4,28,31] and a spatial slider-crank mechanism [3,46,47]. The degrees of freedom are described by means of seven generalized coordinates for each body (three for describing the position of the center of mass and four Euler parameters for describing the attitude), which leads to a singular mass matrix.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Cuadrado et al [29] developed a more efficient implementation of the mass-orthogonal projection which requires only successive forward reductions and back-substitutions, and then Blajer [30] gave a correcting formulation which doesn't need to update the Lagrange multipliers. The energy consideration with velocity projection was studied by Orden et al [31,32], providing an alternative interpretation of its effect on the stability and a practical criterion for the mass-orthogonal projection matrix selection. Blajer [30] also gave a geometrical interpretation to the augmented Lagrangian formulation which is capable of treating systems with changing topologies, redundant constraints, singular positions and some singularity of the mass matrix where the mass-orthogonal projection method can still be applied but the geometric elimination method cannot [20,21,23,24].…”

Section: Introductionmentioning

confidence: 99%

“…(2.18) and (2.19) in [36]. It had been interpreted by Blajer that the mass-orthogonal projection method can deal with some singularity of the mass matrix [28][29][30][31][32], and it is also capable of treating systems with changing topologies, redundant constraints and singular positions which are not considered in this paper. Although we have concluded that large penalty values adversely affect the numerical conditioning of the algebraic linear system and the mass-orthogonal method shows less physical meaning than the geometrical projection method with a mass weight matrix, it had been shown that the leading matrix applied in the mass-orthogonal projection method, which is derived from the augmented Lagrangian formulation [13,14], is always positive definite, i.e., invertible, even in singular positions and/or with linearly dependent constraints and some singularity of the mass matrix.…”

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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Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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“…An approach proposed by Bauchau [4] deals with high frequencies in the multibody system with a self-stabilization algorithm which enforces the constraints in a multibody system in an index-2 formulation similar to the GGL method [11], but avoiding the introduction of additional Lagrange multipliers. Another approach concerning the stabilization of constrained mechanical systems has been presented by García Orden [10], in which the artificial energy introduced in the system can be controlled from various points of view. The group around Arnold, Cardona and Brüls [2,3] suggests a Lie group time integration of constrained systems and presents a stabilized index-2 formulation in terms of Euler parameters; see e.g.…”

Section: Introductionmentioning

confidence: 99%

A modified HHT method for the numerical simulation of rigid body rotations with Euler parameters

et al. 2019

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In multibody dynamics, the Euler parameters are often used for the numerical simulation of rigid body rotations because they lead to a relatively simple form of the rotation matrix which avoids the evaluation of trigonometric functions and can thus save computational time. The Newmark method and the closely related Hilber-Hughes-Taylor (HHT) method are widely employed for solving the equations of motion of mechanical systems. They can also be applied to constrained systems described by differential algebraic equations. However, in the classical versions, the use of these integration schemes have a very unfavorable impact on the Euler parameter description of rotational motions. In this paper, we show analytically that the angular velocity for a rotation about a single axis under a constant moment will not increase linearly but grows slower. This effect, which does not appear for Euler angles, can be even observed if the numerical damping parameter α in the HHT method is set to zero. To circumvent this problem without losing the advantage of Euler parameters, we present a modified HHT method which reduces the damping effect on the angular velocity significantly and eliminates it completely for α = 0.

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“…It represents the extension of the formulations presented in [6,13] to nonholonomic systems and their generalization to more advanced integration schemes. This formulation constitutes a good exponent of what it should be expected from a modern multibody method: it is efficient, accurate, and robust; it exactly satisfies the constraints at position, velocity, and acceleration levels; it has predictable and controllable energy decaying properties [26,24,25]; it can deal with redundant constraints and singular configurations without special considerations [29] and it can handle both holonomic (scleronomic and rheonomic) and nonholonomic constraints.…”

Section: Introductionmentioning

confidence: 99%

Direct sensitivity analysis of multibody systems with holonomic and nonholonomic constraints via an index-3 augmented Lagrangian formulation with projections

et al. 2018

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Optimizing the dynamic response of mechanical systems is often a necessary step during the early stages of product development cycle. This is a complex problem that requires to carry out the sensitivity analysis of the system dynamics equations if gradient-based optimization tools are used. These dynamics equations are often expressed as a highly nonlinear system of Ordinary Differential Equations (ODEs) or Differential-Algebraic Equations (DAEs), if a dependent set of generalized coordinates with its corresponding kinematic constraints is used to describe the motion. Two main techniques are currently available to perform the sensitivity analysis of a multibody system, namely the direct differentiation and the adjoint variable methods.In this paper, we derive the equations that correspond to the direct sensitivity analysis of the index-3 augmented Lagrangian formulation with velocity and acceleration projections. Mechanical systems with both holonomic and nonholonomic constraints are considered. The evaluation of the system sensitivities requires the solution of a Tangent Linear Model (TLM) that corresponds to the Newton-Raphson iterative solution of the dynamics at configuration level, plus two additional nonlinear systems of equations for the velocity and acceleration projections. The method was validated in the sensitivity analysis of a set of examples, including a five-bar linkage with spring elements, which had been used in the literature as 1 D. Dopico et al.benchmark problem for similar multibody dynamics formulations, a point-mass system subjected to nonholonomic constraints, and a full-scale vehicle model.

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Energy considerations for the stabilization of constrained mechanical systems with velocity projection

Cited by 16 publications

References 19 publications

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

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

A modified HHT method for the numerical simulation of rigid body rotations with Euler parameters

Direct sensitivity analysis of multibody systems with holonomic and nonholonomic constraints via an index-3 augmented Lagrangian formulation with projections

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