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.     

Discussion on the optimization problem formulation of flexible components in multibody systems

This paper is dedicated to the structural optimization of flexible components in mechanical systems modeled as multibody systems. While most of the structural optimization developments have been conducted under (quasi-)static loadings or vibration design criteria, the proposed approach aims at considering as precisely as possible the effects of dynamic loading under service conditions. Solving this problem is quite challenging and naive implementations may lead to inaccurate and unstable results. To elaborate a robust and reliable approach, the optimization problem formulation is investigated because it turns out that it is a critical point. Different optimization algorithms are also tested. To explain the efficiency of the various solution approaches, the complex nature of the design space is analyzed. Numerical applications considering the optimization of a two-arm robot subject to a trajectory tracking constraint and the optimization of a slider-crank mechanism with a cyclic dynamic loading are presented to illustrate the different concepts.     

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.     

Non-linear dynamics of flexible multibody systems

In this work we set to examine several important issues pertinent to currently very active research area of the finite element modeling of flexible multibody system dynamics. To that end, we first briefly introduce three different model problems in non-linear dynamics of flexible 3D solid, a rigid body and 3D geometrically exact beam, which covers the vast majority of representative models for the particular components of a multibody system. The finite element semi-discretization for these models is presented along with the time-discretization performed by the mid-point scheme. In extending the proposed methodology to modeling of flexible multibody systems, we also present how to build a systematic representation of any kind of joint connecting two multibody components, a typical case of holonomic contraint, as a linear superposition of elementary constraints. We also indicate by a chosen model of rolling contact, an example of non-holonomic constraint, that the latter can also be included within the proposed framework. An important aspect regarding the reduction of computational cost while retaining the consistency of the model is also addressed in terms of systematic use of the rigid component hypothesis, mass lumping and the appropriate application of the explicit-implicit time-integration scheme to the problem on hand. Several numerical simulations dealing with non-linear dynamics of flexible multibody systems undergoing large overall motion are presented to further illustrate the potential of presented methodology. Closing remarks are given to summarize the recent achievements and point out several directions for future research.     

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.     

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     

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.     

Nonlinear dynamic response structural optimization using equivalent static loads

It is well known that nonlinear dynamic response optimization using a conventional optimization algorithm is fairly difficult and expensive for the gradient or non-gradient based optimization methods because many nonlinear dynamic analyses are required. Therefore, it is quite difficult to find practical large scale examples with many design variables and constraints for nonlinear dynamic response structural optimization. The equivalent static loads (ESLs) method is newly proposed and applied to nonlinear dynamic response optimization. The equivalent static loads are defined as the linear static load sets which generate the same response field in linear static analysis as that from nonlinear dynamic analysis. The ESLs are made from the results of nonlinear dynamic analysis and used as external forces in linear static response optimization. Then the same response from nonlinear dynamic analysis can be considered throughout linear static response optimization. The updated design from linear response optimization is used again in nonlinear dynamic analysis and the process proceeds in a cyclic manner until the convergence criteria are satisfied. Several examples are solved to validate the method. The results are compared to those of the conventional method with sensitivity analysis using the finite difference method.     

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.     

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.     

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...     

Technical overview of the equivalent static loads method for non-linear static response structural optimization

Linear static response structural optimization has been developed fairly well by using the finite element method for linear static analysis. However, development is extremely slow for structural optimization where a non linear static analysis technique is required. Optimization methods using equivalent static loads (ESLs) have been proposed to solve various structural optimization disciplines. The disciplines include linear dynamic response optimization, structural optimization for multi-body dynamic systems, structural optimization for flexible multi-body dynamic systems, nonlinear static response optimization and nonlinear dynamic response optimization. The ESL is defined as the static load that generates the same displacement field by an analysis which is not linear static. An analysis that is not linear static is carried out to evaluate the displacement field. ESLs are evaluated from the displacement field, linear static response optimization is performed by using the ESLs, and the design is updated. This process proceeds in a cyclic manner. A variety of problems have been solved by the ESLs methods. In this paper, the methods are completely overviewed. Various case studies are demonstrated and future research of the methods is discussed.     

Finite element analysis of flexible multibody systems with fuzzy parameters

A computational procedure is presented for predicting the dynamic response and evaluating the sensitivity coefficients of flexible multibody systems whose characteristics include fuzzy parameters. Time-histories of the possibility distributions of the response and the sensitivity coefficients are generated. These coefficients measure the sensitivity of the dynamic response to variations in the material, geometric and external force parameters of the system. The five key components of the procedure are: a) a corotational frame approach used in conjunction with a total Lagrangian formulation; b) beam and shell elements with the Cartesian coordinates of the nodes selected as degrees of freedom, and with continuous inter-element slopes; c) use of an approximate method of extension, based on the α-cut representation, called the ‘vertex method’ for generating the possibility distributions of the desired response quantities and their sensitivity coefficients; d) semi-explicit temporal integration technique for generating the dynamic response; and e) direct differentiation approach for evaluating the sensitivity coefficients. The effectiveness of the procedure and the usefulness of the fuzzy output are demonstrated through numerical examples, including an articulated space structure consisting of beams, shells and revolute joints.     

Three-dimensional formulation of rigid-flexible multibody systems with flexible beam elements

Multibody systems generally contain solids with appreciable deformations and which decisively influence the dynamics of the system. These solids have to be modeled by means of special formulations for flexible solids. At the same time, other solids are of such a high stiffness that they may be considered rigid, which simplifies their modeling. For these reasons, for a rigid-flexible multibody system, two types of formulations coexist in the equations of the system. Among the different possibilities provided in the literature on the material, the formulation in natural coordinates and the formulation in absolute nodal coordinates are utilized in this paper to model the rigid and flexible solids, respectively. This paper contains a mixed formulation based on the possibility of sharing coordinates between a rigid solid and a flexible solid. The global mass matrix of the system is shown to be constant and, in addition, many of the constraint equations obtained upon utilizing these formulations are linear and can be eliminated.     

A nonlinear two-node superelement for use in flexible multibody systems

A nonlinear two-node superelement is proposed for the modeling of flexible complex-shaped links for use in multibody simulations. Assuming that the elastic deformations with respect to a corotational reference frame remain small, substructuring methods may be used to obtain reduced mass and stiffness matrices from a linear finite element model. These matrices are used in the derivation of potential and kinetic energy expressions of the nonlinear two-node superelement. By evaluating Lagrange’s equations, expressions for the internal and external forces acting on the superelement can be obtained. The inertia forces of the superelement are derived in terms of absolute nodal velocities and accelerations, which greatly simplifies the dynamic formulation. Three examples are included. The first two examples are used to validate the method by comparing the results with those obtained from nonlinear beam element solutions. We consider a benchmark simulation of the spin-up motion of a flexible beam with uniform cross-section and a similar simulation in which the beam is simultaneously excited in the out-of-plane direction. Results from both examples show good agreement with simulation results obtained using nonlinear finite beam elements. In a third example, the method is applied to an unbalanced rotating shaft, illustrating the potential of the proposed methodology for a more complex geometry.