On the use of moment-matching to build reduced order models in flexible multibody dynamics

An important issue in the field of flexible multibody dynamics is the reduction of the flexible body's degrees of freedom. For this purpose, often modal reduction through projection onto a subspace spanned by some dominant eigenvectors is used. However, as in this method the dynamical boundary conditions are not taken into account, a large number of eigenmodes is required to obtain a good approximation and also the selection of the dominant modes can be quite difficult. Therefore, the authors propose an approach based on accounting for the flexible body as an input-output system in the frequency domain. The reduced order model is generated by imposing a set of interpolation conditions concerning the values and derivatives of the system's transfer function in a predefined frequency range. This procedure is known as moment-matching and can be realised through projection onto so-called Krylov-subspaces. As this technique allows the incorporation of the frequency content and the spatial distribution of the loads, in the chosen frequency range more accurate reduced order models can be obtained compared to other model reduction techniques available in structural mechanics. The calculation of the Krylov-subspaces can be implemented very efficiently, using the Arnoldi or Lanczos procedure in connection with sparse matrix techniques. The capability of the proposed technique is demonstrated by means of a numerical example.     

A method for improving dynamic solutions in flexible multibody dynamics

This paper presents a method for improving dynamic solutions that are obtained from the dynamic simulation of flexible multibody systems. The mode-acceleration concept in linear structural dynamics is utilized in the proposed method for improving accuracy in the postprocessing stage. A theoretical explanation is made on why the proposed method improves the dynamic solutions in the context of the mode-acceleration method. A mode-acceleration equation for each flexible body is defined and the load term in the right hand side of the equation is represented as a combination of space-dependent and time-dependent terms so that efficient computation of dynamic solutions can be achieved. The load term is obtained from dynamic simulation of a flexible multibody system and a finite element method is used to compute dynamic solutions by quasi-static analyses. Numerical examples show the effectiveness of the proposed method.     

Frequency-domain generelaized singular peruturbation method for relative error model order reduction

A new mixed method for relative error model order reduction is proposed. In the proposed method the frequency domain balanced stochastic truncation method is improved by applying the generalized singular perturbation method to the frequency domain balanced system in the reduction procedure. The frequency domain balanced stochastic truncation method, which was proposed in [15] and [17] by the author, is based on two recently developed methods, namely frequency domain balanced truncation within a desired frequency bound and inner-outer factorization techniques. The proposed method in ttiis paper is a carry over of the frequency-domain balanced stochastic truncation and is of interest for practical model order reduction because in this context it shows to keep the accuracy of the approximation as high as possible without sacrificing the computational efficiency and important system properties. It is shown that some important properties of the frequency domain stochastic balanced reduction technique are extended to the proposed reduction method by using the concept and properties of the reciprocal systems. Numerical results show the accuracy, simplicity and flexibility enhancement of the method.     

Stability evaluation and system identification of flexible multibody systems

This paper is concerned with the linearized stability analysis and system identification of flexible multibody systems. Two closely related stability analysis approaches are summarized. Next, these approaches are extended to provide robust system identification procedures that combine least squares techniques and Kalman filters. The singular value decomposition, a numerically stable mathematical tool, is used to improve the robustness of the algorithm. The proposed algorithm identifies a minimum order plant based on input-output data, and is applicable to both experimental measurements or numerically computed responses. The proposed approaches are computationally inexpensive and consist of purely post processing steps that can be used with any multi-physics computational multibody tool or with experimental data. Commemorative Contribution.     

The development of a sliding joint for very flexible multibody dynamics using absolute nodal coordinate formulation

In this paper, a formulation for a spatial sliding joint is derived using absolute nodal coordinates and non-generalized coordinate and it allows a general multibody move along a very flexible cable. The large deformable motion of a spatial cable is presented using absolute nodal coordinate formulation, which is based on the finite element procedures and the general continuum mechanics theory to represent the elastic forces. And the nongeneralized coordinate, which is related to neither the inertia forces nor the external forces, is used to describe an arbitrary position along the centerline of a very flexible cable. Hereby, the non-generalized coordinate represents the arc-length parameter. The constraint equations for the sliding joint are expressed in terms of generalized coordinate and nongeneralized coordinate. In the constraint equations for the sliding joint, one constraint equation can be systematically eliminated. There are two independent Lagrange multipliers in the final system equations of motion associated with the sliding joint. The development of this sliding joint is important to analyze many mechanical systems such as pulley systems and pantograph-catenary systems for high speed-trains.     

Model decomposition and reduction tools for large-scale networks in systems biology

Biological system models are routinely developed in modern systems biology research following appropriate modelling/experiment design cycles. Frequently these take the form of high-dimensional nonlinear Ordinary Differential Equations that integrate information from several sources; they usually contain multiple time-scales making them difficult even to simulate. These features make systems analysis (understanding of robust functionality) — or redesign (proposing modifications in order to improve or modify existing functionality) a particularly hard problem. In this paper we use concepts from systems theory to develop two complementary tools that can help us understand the complex behaviour of such system models: one based on model decomposition and one on model reduction. Our aim is to algorithmically produce biologically meaningful, simplified models, which can then be used for further analysis and design. The tools presented are applied on a model of the Epidermal Growth Factor signalling pathway.     

Composite rigid body formalism for flexible multibody systems

The paper describes the extension of the composite rigid body formalism for the flexible multibody systems. The extension has been done in such a way that all advantages of the formalism with respect to the coordinates of large motion of rigid bodies are extended to the flexible degrees of freedom, e.g. the same recursive treatment of both coordinates and no appearance of O(n ³) computational complexity terms due to the flexibility. This extension has been derived for both open loop and closed loop systems of flexible bodies. The comparison of the computational complexity of this formalism with other known approaches has shown that the described formalism of composite rigid body and the residual algorithm based on it are more efficient formalisms for small number of bodies in the chains and deformation modes than the usual recursive formalism of articulated body inertia.     

多柔体系统动力学研究进展与挑战

首先回顾多体系统动力学的学科发展和学术交流情况,然后系统概述了多柔体系统动力学方程数值算法、多柔体系统接触/碰撞动力学与柔性空间结构展开动力学三个方面的研究进展及值得关注的若干问题,最后给出了开展多柔体系统动力学研究的若干建议.     

Generalised gramian framework for model/controller order reduction of switched systems

In this article, a general method for model/controller order reduction of switched linear dynamical systems is presented. The proposed technique is based on the generalised gramian framework for model reduction. It is shown that different classical reduction methods can be developed into a generalised gramian framework. Balanced reduction within a specified frequency bound is developed within this framework. In order to avoid numerical instability and also to increase the numerical efficiency, generalised gramian‐based Petrov–Galerkin projection is constructed instead of the similarity transform approach for reduction. The framework is developed for switched controller reduction. To the best of our knowledge, there is no other reported result on switched controller reduction in the literature. The method preserves the stability under an arbitrary switching signal for both model and controller reduction. Furthermore, it is applicable to both continuous and discrete time systems for different classical gramian‐based reduction methods. The performance of the proposed method is illustrated by numerical examples.     

Analysis of dynamic strains in tibia during human locomotion based on flexible multibody approach integrated with magnetic resonance imaging technique

Bone is known to adapt to the prevalent strain environment while the variation in strains, e.g., due to mechanical loading, modulates bone remodeling, and modeling. Dynamic strains rather than static strains provide the primary stimulus of bone functional adaptation. The finite element method can be generally used for estimating bone strains, but it may be limited to the static analysis of bone strains since the dynamic analysis requires expensive computation. Direct in vivo strain measurement, in turn, is an invasive procedure, limited to certain superficial bone sites, and requires surgical implementation of strain gauges and thus involves risks (e.g., infection). Therefore, to overcome difficulties associated with the finite element method and the in vivo strain measurements, the flexible multibody simulation approach has been recently introduced as a feasible method to estimate dynamic bone strains during physical activity. The purpose of the present study is to further strengthen the idea of using the flexible multibody approach for the analysis of dynamic bone strains. Besides discussing the background theory, magnetic resonance imaging is integrated into the flexible multibody approach framework so that the actual bone geometry could be better accounted for and the accuracy of prediction improved.     

基于等几何分析的柔性多体系统建模方法研究

近三十年来,柔性多体系统动力学取得长足进步,尤其是以绝对节点坐标方法(Absolute Nodal Coordinate Formulation, ANCF)为代表的非线性有限元已被用来处理复杂的柔性多体系统动力学问题.但绝对节点坐标方法采用斜率矢量作为广义坐标,导致系统自由度多,计算效率低.针对柔性多体系统,基于非均匀有理B样条(Non-Uniform Rational B-Splines, NURBS)曲线和曲面分别提出了Euler-Bernoulli细长梁单元和Kirchhoff-Love薄壳单元,在完全拉格朗日格式下,根据Green应变张量对单元变形进行描述,结合第二类Piola-Kirchhoff应力张量给出单元应变能公式,推导了单元的弹性力和弹性力雅可比矩阵表达式,最后通过静力学及动力学数值算例对提出的两类单元的性能进行对比和验证,为柔性多体系统建模提供了一种精确高效的新单元.     

Identification and data-driven model reduction of state-space representations of lossless and dissipative systems from noise-free data

We illustrate procedures to identify a state-space representation of a lossless or dissipative system from a given noise-free trajectory; important special cases are passive systems and bounded-real systems. Computing a rank-revealing factorization of a Gramian-like matrix constructed from the data, a state sequence can be obtained; the state-space equations are then computed by solving a system of linear equations. This idea is also applied to perform model reduction by obtaining a balanced realization directly from data and truncating it to obtain a reduced-order model.     

Coprime factor model reduction for discrete-time uncertain systems

This paper presents a contractive coprime factor model reduction approach for discrete-time uncertain systems of LFT form with norm bounded structured uncertainty. A systematic approach is proposed for coprime factorization and contractive coprime factorization of the underlying uncertain systems. The proposed coprime factor approach overcomes the robust stability restriction on the underlying systems which is required in the balanced truncation approach. Our method is based on the use of LMIs to construct the desired reduced dimension uncertain system model. Closed-loop robustness is discussed under additive coprime factor perturbations.     

Optimization of a complex flexible multibody systems with composite materials

The paper presents a general optimization methodology for flexible multibody systems which is demonstrated to find optimal layouts of fiber composite structures components. The goal of the optimization process is to minimize the structural deformation and, simultaneously, to fulfill a set of multidisciplinary constraints, by finding the optimal values for the fiber orientation of composite structures. In this work, a general formulation for the computation of the first order analytical sensitivities based on the use of automatic differentiation tools is applied. A critical overview on the use of the sensitivities obtained by automatic differentiation against analytical sensitivities derived and implemented by hand is made with the purpose of identifying shortcomings and proposing solutions. The equations of motion and sensitivities of the flexible multibody system are solved simultaneously being the accelerations and velocities of the system and the sensitivities of the accelerations and of the velocities integrated in time using a multi-step multi-order integration algorithm. Then, the optimal design of the flexible multibody system is formulated to minimize the deformation energy of the system subjected to a set of technological and functional constraints. The methodologies proposed are first discussed for a simple demonstrative example and applied after to the optimization of a complex flexible multibody system, represented by a satellite antenna that is unfolded from its launching configuration to its functional state.     

Task-level approaches for the control of constrained multibody systems

This paper presents a task-level control methodology for the general class of holonomically constrained multibody systems. As a point of departure, the general formulation of constrained dynamical systems is reviewed with respect to multiplier and minimization approaches. Subsequently, the operational space framework is considered and the underlying symmetry between constrained dynamics and operational space control is discussed. Motivated by this symmetry, approaches for constrained task-level control are presented which cast the general formulation of constrained multibody systems into a task space setting using the operational space framework. This provides a means of exploiting task-level control structures, native to operational space control, within the context of constrained systems. This allows us to naturally synthesize dynamic compensation for a multibody system, that properly accounts for the system constraints while performing a control task. A set of examples illustrate this control implementation. Additionally, the inclusion of flexible bodies in this approach is addressed.     

A global piecewise smooth Newton method for fast large-scale model predictive control

In this paper, the strictly convex quadratic program (QP) arising in model predictive control (MPC) for constrained linear systems is reformulated as a system of piecewise affine equations. A regularized piecewise smooth Newton method with exact line search on a convex, differentiable, piecewise-quadratic merit function is proposed for the solution of the reformulated problem. The algorithm has considerable merits when applied to MPC over standard active set or interior point algorithms. Its performance is tested and compared against state-of-the-art QP solvers on a series of benchmark problems. The proposed algorithm is orders of magnitudes faster, especially for large-scale problems and long horizons. For example, for the challenging crude distillation unit model of Pannocchia, Rawlings, and Wright (2007) with 252 states, 32 inputs, and 90 outputs, the average running time of the proposed approach is 1.57 ms.