Linear dynamics of flexible multibody systems

We consider mechanical systems where the dynamics are partially constrained to prescribed trajectories. An example for such a system is a building crane with a load and the requirement that the load moves on a certain path.Enforcing this condition directly in form of a servo constraint leads to differential-algebraic equations (DAEs) of arbitrarily high index. Typically, the model equations are of index 5, which already poses high regularity conditions. If we relax the servo constraints and consider the system from an optimal control point of view, the strong regularity conditions vanish, and the solution can be obtained by standard techniques.By means of the well-known \(n\)-car example and an overhead crane, the theoretical and expected numerical difficulties of the direct DAE and the alternative modeling approach are illustrated. We show how the formulation of the problem in an optimal control context works and address the solvability of the optimal control system. We discuss that the problematic DAE behavior is still inherent in the optimal control system and show how its evidences depend on the regularization parameters of the optimization.     

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.     

Port-Hamiltonian flexible multibody dynamics

Multibody System Dynamics - A new formulation for the modular construction of flexible multibody systems is presented. By rearranging the equations for a flexible floating body and introducing the...     

Pulse control of flexible multibody systems

An active pulse control method is developed to reduce the vibrations of multibody systems resulting from impact loadings. The pulse, which is a function of system generalized coordinates and velocities, is determined analytically using energy and momentum balance equations of the impacting bodies. Elastic components in the multibody system are discretized using the finite element method. The system equations of motions and nonlinear algebraic constraint equations describing mechanical joints between different components are written in the Lagrangian formulation using a finite set of coupled reference position and local elastic generalized coordinates. A set of independent differential equations are identified by the generalized coordinate partitioning of the constraint Jacobian matrix. These equations are written in the state space formulation and integrated forward in time using a direct numerical integration method. Dependent coordinates are then determined using the constraint kinematic relations. Points in time at which impact occurs are monitored by an impact predictor function, which controls the integration algorithms and forces for the solution of the momentum relation, to define the jump discontinuities in the composite velocity vector as well as the system reaction forces. The effectiveness of the active pulse control in reducing the vibration of flexible multibody aircraft during the touchdown impact is investigated and numerical results are presented.     

Galerkin meshfree methods applied to the nonlinear dynamics of flexible multibody systems

Meshfree Galerkin methods have been developed recently for the simulation of complex mechanical problems involving large strains of structures, crack propagation, or high velocity impact dynamics. At the present time, the application of these methods to multibody dynamics has not been made despite their great advantage in some situations over standard finite element techniques.     

Isogeometric shell discretizations for flexible multibody dynamics

This work aims at including nonlinear elastic shell models in a multibody framework. We focus our attention to Kirchhoff–Love shells and explore the benefits of an isogeometric approach, the latest development in finite element methods, within a multibody system. Isogeometric analysis extends isoparameteric finite elements to more general functions such as B-splines and NURBS (Non-Uniform Rational B-Splines) and works on exact geometry representations even at the coarsest level of discretizations. Using NURBS as basis functions, high regularity requirements of the shell model, which are difficult to achieve with standard finite elements, are easily fulfilled. A particular advantage is the promise of simplifying the mesh generation step, and mesh refinement is easily performed by eliminating the need for communication with the geometry representation in a CAD (Computer-Aided Design) tool. Target applications are wind turbine blades and twist beam rear suspensions. First numerical examples demonstrate an impressive convergence behavior of the isogeometric approach even for a coarse mesh, while offering substantial savings with respect to the number of degrees of freedom.     

Modal analysis of active flexible multibody systems

When performing modal analyses of active flexible multibody systems, both controller effects and flexible body dynamics should be included in a multidisciplinary system model. This paper deals with the theory of solving the closed-loop eigenvalue problem for active flexible multibody systems with multiple-input multiple-output proportional-integral-derivative (PID) type feedback controllers and multiple degrees of freedom finite element models. Modal analyses are performed on both a simple and complex active flexible multibody system in order to illustrate the difference between current modal analysis method for such systems and the proposed theory derived in this paper.     

Passivity-based control of nonlinear flexible multibody systems

In this paper, global asymptotic stability of a class of nonlinear multibody flexible space structures under certain dissipative compensation is established. Furthermore, for an important subclass of such systems, the stability is shown to be robust to certain types of actuator and sensor nonlinearities. The results are applicable to robust stabilization of a wide class of systems, including flexible space structures and manipulators with articulated flexible appendages. The stability proofs use the Lyapunov approach and exploit the inherent passivity of such systems     

Application of perturbation techniques to flexible multibody system dynamics

In this paper, a matrix perturbation technique is developed for flexible bodies (substructures) that undergo large reference translational and rotational displacements. Although the governing dynamic equations of motion of such systems are highly nonlinear because of the large angular rotations and the resulting nonlinear inertia coupling between the reference motion and the elastic deformation, a generalized linear eigenvalue problem that defines the deformation mode shapes of the body with respect to the selected body reference is identified. This eigenvalue problem is solved only once and the variations in the body stiffness and inertia properties due to a change in selected design parameters are evaluated by using perturbation analysis techniques. The main advantage of using the proposed technique is to avoid a new finite element discretization when some design parameters are changed. This, in turn, substantially reduces the computational time, especially when large scale flexible bodies with complex geometry are considered. A numerical example is presented in order to demonstrate the use of the perturbation techniques developed in this paper in the design of flexible multibody systems.     

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.     

Object-oriented virtual environment for visualization of flexible multibody systems

An object-oriented event-driven virtual environment (VE) for viewing the simulation results of flexible multibody systems (FMS) is developed. The VE interfaces with the following output devices: immersive stereoscopic screen(s) and stereo speakers; and a variety of input devices including, head tracker, wand, joystick, mouse, microphone, and keyboard. The VE incorporates the following types of primitive software objects: user-interface objects, support objects, geometric entities, and finite elements. Each object encapsulates a set of properties, methods, and events that define its behavior, appearance, and functions. A “container” object allows grouping many objects into one object, which inherits the properties of its “children” objects. The VE allows real-time viewing and “fly-through” of photo-realistic models, vibrational mode-shapes, and animation of the dynamic motion of FMS. An application of this VE is presented for visualization of the dynamic analysis results of a large deployable space structure — NASA’s Next Generation Space Telescope.     

Simulation and stability analysis of periodic flexible multibody systems

Multibody System Dynamics - The dynamic response of many flexible multibody systems of practical interest is periodic. The investigation of such problems involves two intertwined tasks: first, the...     

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.     

Application of deformable-body mean axis to flexible multibody system dynamics

While a body-fixed axis is commonly employed as a reference for rigid components, a moving axis is suggested for flexible bodies that undergo large rotations. There are, however, an infinite number of arrangements for the body axes of flexible components. In order to define a unique displacement field, a set of reference conditions that define the nature of the body axis have to be imposed. These reference conditions are mainly dictated by physical considerations. In multibody systems the motion of flexible bodies are constrained by and coupled with the motion of other components in the system through a set of nonlinear constraint equations that describe joints between different bodies. In this paper the problem of selecting an axis for flexible bodies is addressed in the frame of a general finite element development for both planar and spatial multibody systems. A body reference satisfying the mean-axis conditions has been selected for a detailed discussion.     

Global modes for the reduction of flexible multibody systems

Multibody System Dynamics - Modeling a flexible multibody system employing the floating frame of reference formulation (FFRF) requires significant computational resources when the flexible...     

Validation of flexible multibody dynamics beam formulations using benchmark problems

As the need to model flexibility arose in multibody dynamics, the floating frame of reference formulation was developed, but this approach can yield inaccurate results when elastic displacements becomes large. While the use of three-dimensional finite element formulations overcomes this problem, the associated computational cost is overwhelming. Consequently, beam models, which are one-dimensional approximations of three-dimensional elasticity, have become the workhorse of many flexible multibody dynamics codes. Numerous beam formulations have been proposed, such as the geometrically exact beam formulation or the absolute nodal coordinate formulation, to name just two. New solution strategies have been investigated as well, including the intrinsic beam formulation or the DAE approach. This paper provides a systematic comparison of these various approaches, which will be assessed by comparing their predictions for four benchmark problems. The first problem is the Princeton beam experiment, a study of the static large displacement and rotation behavior of a simple cantilevered beam under a gravity tip load. The second problem, the four-bar mechanism, focuses on a flexible mechanism involving beams and revolute joints. The third problem investigates the behavior of a beam bent in its plane of greatest flexural rigidity, resulting in lateral buckling when a critical value of the transverse load is reached. The last problem investigates the dynamic stability of a rotating shaft. The predictions of eight independent codes are compared for these four benchmark problems and are found to be in close agreement with each other and with experimental measurements, when available.