Error-Controlled Model Reduction in Flexible Multibody Dynamics

2010 ◽  
Vol 5 (3) ◽  
Author(s):  
Jörg Fehr ◽  
Peter Eberhard
Keyword(s):  
Model Reduction ◽  
Error Bounds ◽  
Degrees Of Freedom ◽  
A Priori ◽  
Reduction Process ◽  
Flexible Multibody Dynamics ◽  
Scanning Tunneling ◽  
Error Estimator ◽  
Gramian Matrix ◽  
Flexible Multibody

One important issue for the simulation of flexible multibody systems is the quality controlled reduction in the flexible bodies degrees of freedom. In this work, the procedure is based on knowledge about the error induced by model reduction. For modal reduction, no error bound is available. For Gramian matrix based reduction methods, analytical error bounds can be developed. However, due to numerical reasons, the dominant eigenvectors of the Gramian matrix have to be approximated. Within this paper, two different methods are presented for this purpose. For moment matching methods, the development of a priori error bounds is still an active field of research. In this paper, an error estimator based on a new second order adaptive global Arnoldi algorithm is introduced and further assists the user in the reduction process. We evaluate and compare those methods by reducing the flexible degrees of freedom of a rack used for active vibration damping of a scanning tunneling microscope.

scholarly journals Iterative Coordinate Reduction Algorithm of Flexible Multibody Dynamics Using a Posteriori Eigenvalue Error Estimation

2020 ◽  
Vol 10 (20) ◽  
pp. 7143
Author(s):  
Seongji Han ◽  
Jin-Gyun Kim ◽  
Juhwan Choi ◽  
Jin Hwan Choi
Keyword(s):  
Multibody Dynamics ◽  
Reduction Process ◽  
Flexible Multibody Dynamics ◽  
Dynamics Simulation ◽  
Iteration Step ◽  
Error Estimator ◽  
A Posteriori ◽  
Flexible Bodies ◽  
Flexible Multibody ◽  
Coordinate Reduction

Coordinate reduction has been widely used for efficient simulation of flexible multibody dynamics. To achieve the reduction of flexible bodies with reasonable accuracy, the appropriate number of dominant modes used for the reduction process must be selected. To handle this issue, an iterative coordinate reduction strategy is introduced. In the iteration step, more dominant modes of flexible bodies are selected than the ones in the previous step. Among the various methods, the conventional frequency cut-off rule is here considered. As a stop criterion, a novel a posteriori error estimator that can evaluate the relative eigenvalue error between full and reduced flexible bodies is proposed. Through the estimated relative eigenvalue error obtained, the number of dominant modes is automatically selected to satisfy the error tolerance up to the desired mode range. The applicability to the automation process is verified through numerical examples. It is also evaluated that efficient and accurate flexible multibody dynamics simulation is available with the reduced flexible body, generated by the proposed algorithm.

Lumped-Parameters Brick Element for Modeling Shell Flexible Multibody Systems

2001 ◽  
Author(s):  
Tamer M. Wasfy
Keyword(s):  
Multibody Dynamics ◽  
Degrees Of Freedom ◽  
Temporal Integration ◽  
Flexible Multibody Dynamics ◽  
Test Problems ◽  
Large Space ◽  
Surface Shear ◽  
Integration Algorithm ◽  
Flexible Multibody ◽  
Brick Element

Abstract An eight-node lumped-parameter brick element, suitable for modeling flexible multibody shell components, is described. The element is composed of twelve truss sub-elements for modeling the membrane and bending modes and six surface shear elements for modeling the shear and warping modes. The element is strategically designed to eliminate locking and spurious modes. Cartesian nodal coordinates are used as degrees-of-freedom with no rotational degrees-of-freedom. This simplifies the coordinate transformations and inertia calculations. A total Lagrangian displacement formulation where the element deformations are measured relative to the unstressed element is employed. The equations of motion are integrated using an explicit temporal integration algorithm. Standard finite element and flexible multibody dynamics test problems are solved to demonstrate the accuracy and robustness of the element. Also, the use of the element in a practical flexible multibody dynamics application, namely, deployment of a large space structure, is demonstrated.

System-Level Modal Representation of Flexible Multibody Systems

2009 ◽  
Author(s):  
Gert H. K. Heirman ◽  
Wim Desmet
Keyword(s):  
Model Reduction ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Flexible Multibody Systems ◽  
System Level ◽  
Algebraic Equations ◽  
Flexible Multibody ◽  
Simulation Results ◽  
Model Equations ◽  
Model Reduction Technique

The presence of both differential and algebraic equations in the model equations, as well as the number of degrees of freedom needed to accurately represent flexibility, prohibit fast simulation of flexible multibody systems (e.g. real-time). In this research, Global Modal Parametrization, a model reduction technique for flexible multibody systems is further developed to speed up simulation of flexible multibody systems. The reduction of the model is achieved by projection on a curvilinear subspace instead of a fixed vector space, requiring significantly less degrees of freedom to represent the system dynamics with the same level of accuracy. The complexity of simulation of the reduced model equations is estimated. In a numerical experiment, simulation results for the original model equations are compared with simulation results for the model equations obtained after model reduction, showing a good match. The dominant sources of error of the proposed methodology are illustrated and explained.

MODEL REDUCTION WITH GEOMETRIC STIFFENING NONLINEARITIES FOR DYNAMIC SIMULATIONS OF MULTIBODY SYSTEMS

2013 ◽  
Vol 13 (08) ◽  
pp. 1350046 ◽  
Author(s):  
FENGXIA WANG
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Model Reduction ◽  
Reduction Process ◽  
Flexible Multibody Dynamics ◽  
Element Model ◽  
Reduced Model ◽  
Matrix Operation ◽  
Reduced Models ◽  
Geometric Stiffening

This work investigates the implementation of nonlinear model reduction to flexible multibody dynamics. Linear elastic theory will lead to instability issues with rotating beam-like structures, due to the neglecting of the membrane-bending coupling on the beam cross-section. During the past decade, considerable efforts have been focused on the derivation of geometric nonlinear formulation based on nodal coordinates. In order to reduce the computation cost in flexible multibody dynamics, which is extremely important for complex large system simulations, modal reduction is usually implemented to a rotating flexible structure with geometric nonlinearities retained in the model. In this work, a standard model reduction process based on matrix operation is developed, and the essential geometric stiffening nonlinearities are retained in the reduced model. The time responses of a tip point on a rotating Euler–Bernoulli blade are calculated based on two nonlinear reduced models. The first reduced model is derived by the standard matrix operation from a full finite element model and the second reduced model is obtained via the Galerkin method. The matrix operation model reduction process is validated through the comparison of the simulation results obtained from these two different reduced models. An interesting phenomenon is observed in this work: In the nonlinear model, if only quadratic geometric stiffing term is retained, the two reduced models converge to the full finite element model with only one bending mode and two axial modes. While if both quadratic and cubic geometric stiffing terms are retained in the nonlinear equation, the modal-based reduced model will not converge to the finite element model unless all eigenmodes are retained, that is the reduced model has no degree of freedom reduction at all.

scholarly journals Input–output measures for model reduction and closed-loop control: application to global modes

2011 ◽  
Vol 685 ◽  
pp. 23-53 ◽  
Author(s):  
Alexandre Barbagallo ◽  
Denis Sipp ◽  
Peter J. Schmid
Keyword(s):  
Feedback Control ◽  
Model Reduction ◽  
Degrees Of Freedom ◽  
Closed Loop ◽  
Reduction Process ◽  
Open Loop ◽  
Input Output ◽  
Control Applications ◽  
Reduced Order ◽  
Global Modes

AbstractFeedback control applications for flows with a large number of degrees of freedom require the reduction of the full flow model to a system with significantly fewer degrees of freedom. This model-reduction process is accomplished by Galerkin projections using a reduction basis composed of modal structures that ideally preserve the input–output behaviour between actuators and sensors and ultimately result in a stabilized compensated system. In this study, global modes are critically assessed as to their suitability as a reduction basis, and the globally unstable, two-dimensional flow over an open cavity is used as a test case. Four criteria are introduced to select from the global spectrum the modes that are included in the reduction basis. Based on these criteria, four reduced-order models are tested by computing open-loop (transfer function) and closed-loop (stability) characteristics. Even though weak global instabilities can be suppressed, the concept of reduced-order compensators based on global modes does not demonstrate sufficient robustness to be recommended as a suitable choice for model reduction in feedback control applications. The investigation also reveals a compelling link between frequency-restricted input–output measures of open-loop behaviour and closed-loop performance, which suggests the departure from mathematically motivated ${\mathscr{H}}_{\infty } $-measures for model reduction toward more physically based norms; a particular frequency-restricted input–output measure is proposed in this study which more accurately predicts the closed-loop behaviour of the reduced-order model and yields a stable compensated system with a markedly reduced number of degrees of freedom.

Three-Dimensional Non-Linear Shell Theory for Flexible Multibody Dynamics

2015 ◽  
Author(s):  
S. L. Han ◽  
O. A. Bauchau
Keyword(s):  
Shell Theory ◽  
Three Dimensional ◽  
Reduction Process ◽  
Flexible Multibody Dynamics ◽  
Stress And Strain ◽  
Two Dimensional ◽  
Governing Equations ◽  
Strain Fields ◽  
Flexible Multibody ◽  
Material Line

In flexible multibody systems, many components are approximated as shells. Classical shell theories, such as Kirchhoff or Reissner-Mindlin shell theory, form the basis of the analytical development for shell dynamics. While such approach is capable of capturing the kinetic energy of the system accurately, it cannot represent the strain energy adequately. For instance, it is well known from three-dimensional elasticity theory that the normal material line will warp under load for laminated composite shells, leading to three-dimensional deformations that generate complex stress states. To overcome this problem, a novel three-dimensional shell theory is proposed in this paper. Kinematically, the problem is decomposed into an arbitrarily large rigid-normal-material-line motion and a warping field. The sectional strains associated with the rigid-normal-material-line motion and the warping field are assumed to remain small. As a consequence of this kinematic decomposition, the governing equations of the problem fall into two distinct categories: the global equations describing geometrically exact shells and the local equations describing local deformations. The governing equations for geometrically exact shells are nonlinear, two-dimensional equations, whereas the local equations are linear, one dimensional, provide the detailed distribution of three-dimensional stress and strain fields. Based on a set of approximated solutions, the local equations is reduced to the corresponding global equations. In the reduction process, a 9 × 9 sectional stiffness matrix can be found, which takes into account the warping effects due to material heterogeneity. In the recovery process, three-dimensional stress and strain fields at any point in the shell can be recovered from the two-dimensional shell solution. The proposed method proposed is valid for anisotropic shells with arbitrarily complex through-the-thickness lay-up configuration.

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