Simulation Of Flexible Multibody Systems Research Articles

A hybrid data-driven model order reduction strategy for flexible multibody systems considering impact and friction

The numerical simulation of flexible multibody systems (FMBSs) with contact is important. Meanwhile, the contact problem is always a key and difficult point for FMBSs; and when impact and friction affect FMBSs, the computational efficiency will dramatically decline. In this work, the nonsmooth contact method (NSCM) for FMBSs is briefly introduced. Moreover, in order to save the computational costs of a contact FMBS, a novel hybrid model order reduction (MOR) strategy based on the idea of data-driven method is proposed. For the hybrid MOR strategy, the reduced-order models (ROMs) based on the NSCM in smooth force part and nonsmooth force part are derived using proper symplectic decomposition and proper orthogonal decomposition, respectively. In addition, in order to adapt the ROMs to parameter variation, the parameter interpolation method on the tangent space of Grassmann manifold is studied. Finally, several numerical examples are tested using the proposed methodology, the results show that compared with the original model, the ROM can save 65.89% of the computing time, and the root-mean-square error between the ROM and original model can reach the order of 10−12.

The tangent stiffness matrix for an absolute interface coordinates floating frame of reference formulation

In this work, a full and complete development of the tangent stiffness matrix is presented, suitable for the use in an absolute interface coordinates floating frame of reference formulation. For simulation of flexible multibody systems, the floating frame formulation is used for its advantage to describe local elastic deformation by means of a body’s linear finite element model. Consequently, the powerful Craig–Bampton method can be applied for model order reduction. By establishing a coordinate transformation from the absolute floating frame coordinates and local interface coordinates corresponding to the Craig–Bampton modes to absolute interface coordinates, it is possible to satisfy kinematic constraints without the use of Lagrange multipliers. In this way, the floating frame does not need to be located at an interface point and can be positioned close to the body’s center of mass, without requiring an interface point at the center of mass. This improves simulation accuracy. In this work, the expression for the new method’s tangent stiffness matrix is obtained by taking the variation of the equation of equilibrium. The global tangent stiffness matrix is expressed as a local tangent stiffness matrix, consisting of both material stiffness and geometric stiffness terms, transformed to the global frame by the rotation matrix of the floating frame. Simulations of static and dynamic validation problems are performed. These simulations show the importance of including the tangent stiffness matrix for both convergence and simulation efficiency.

Discrete Adjoint Method for the Sensitivity Analysis of Flexible Multibody Systems

The gradient-based design optimization of mechanical systems requires robust and efficient sensitivity analysis tools. The adjoint method is regarded as the most efficient semi-analytical method to evaluate sensitivity derivatives for problems involving numerous design parameters and relatively few objective functions. This paper presents a discrete version of the adjoint method based on the generalized-alpha time integration scheme, which is applied to the dynamic simulation of flexible multibody systems. Rather than using an ad hoc backward integration solver, the proposed approach leads to a straightforward algebraic procedure that provides design sensitivities evaluated to machine accuracy. The approach is based on an intrinsic representation of motion that does not require a global parameterization of rotation. Design parameters associated with rigid bodies, kinematic joints, and beam sectional properties are considered. Rigid and flexible mechanical systems are investigated to validate the proposed approach and demonstrate its accuracy, efficiency, and robustness.

Dynamic fracture simulation of flexible multibody systems via coupled finite elements of ANCF and particles of SPH

A parallel computation methodology is proposed to study the dynamic fracture process of a flexible multibody system with initial cracks. The potential fracture domains of the flexible body system are described by using the particles of smoothed particle hydrodynamics (SPH), and the other domains of the system are modeled by using the finite elements of absolute nodal coordinate formulation (ANCF). In order to preserve the continuity of deformation field, extra virtual particles are uniformly embedded into the interface, where the finite elements of ANCF and the particles of SPH are connected, so as to transmit the interaction forces. The OpenACC derivatives are used to parallelize both the particle contact detection and the solution of the integral equations. A predictor-corrector scheme is used to solve the ordinary differential equations for the particles of SPH, while the generalized-alpha method is used to solve the huge set of differential algebraic equations for the multibody system. The OpenMP derivatives are also used to parallelize the evaluation of the elastic force vectors and their Jacobi matrices of the finite elements. Finally, three case studies are given to validate the proposed computation methodology.

Flexible multibody simulation using hybrid integration scheme

One issue in the dynamic simulation of flexible multibody system is poor computation efficiency, which is due to high frequency components in the solution associated with a deformable body. Standard explicit numerical methods should take very small time steps in order to satisfy the absolute stability condition for the high frequency components and, in turn, the computational efficiency deteriorates. In this study, a hybrid integration scheme is applied to solve the equations of motion of a flexible multibody system for achieving better computational efficiency. The computation times and simulation results are compared between the hybrid scheme and conventional methods. The results demonstrate that the efficiency of a flexible multibody simulation can be improved by using the hybrid scheme.

Coupled differential algebraic equations in the simulation of flexible multibody systems with hydrodynamic force elements

AbstractThe mathematical modelling of elastohydrodynamic bearings in combustion engines leads to a coupled system of (partial) differential algebraic equations, which is represented by a flexible multibody system model of the engine including crankshaft and bearing and by the Reynolds equation that describes the non‐linear effects in the fluid film. The hydrodynamic forces depend strongly on the position and the elastic deformation of crankshaft and bearing shell. We discuss the influence of the spatial discretization on accuracy and numerical effort. Since a fine spatial discretization substantially slows down the numerical solution, we propose a semi‐analytical method based on singular perturbation theory to speed‐up time integration. Numerical tests for a simplified benchmark problem illustrate the advantages of this approach. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The Study on 3D Incompatible Element in Flexible Multibody System Dynamics

In this study the performance of three-dimensional tetrahedron elements has been compared with the background of the simulation of Flexible Multibody System (FMBS). The standard linear tetrahedron element has the least number of nodes, however it has the locking problem. The standard quadratic element has shown precise result and robust performance in many applications, but its high nodes also means large computational time. The incompatible elements integrates the excellence of the linear and quadratic elements and its excellent performance in term of accuracy and time efficiency is presented.

Fluid-Conveying Flexible Pipes Modeled by Large-Deflection Finite Elements in Multibody Systems

The modeling and simulation of flexible multibody systems containing fluid-conveying pipes are considered. It is assumed that the mass-flow rate is prescribed and constant and the pipe cross section is piecewise uniform. An existing beam element capable of handling large motions is modified to include the effect of the fluid flow and the initial curvature of the pipe. The modified element is incorporated in a finite-element based multibody system dynamics program, which takes care of the connection with other parts of the system and the simulation. The element is applied in several test problems: the buckling of a simply supported pipe, the flutter instability of a cantilever pipe, and the motion of a curved pipe that can rotate about an axis perpendicular to its plane. As a three-dimensional example, a Coriolis mass-flow rate meter with a U-shaped pipe is considered.

A-posteriori error estimation for second order mechanical systems

One important issue for the simulation of flexible multibody systems is the reduction of the flexible bodies degrees of freedom. As far as safety questions are concerned knowledge about the error introduced by the reduction of the flexible degrees of freedom is helpful and very important. In this work, an a-posteriori error estimator for linear first order systems is extended for error estimation of mechanical second order systems. Due to the special second order structure of mechanical systems, an improvement of the a-posteriori error estimator is achieved. A major advantage of the a-posteriori error estimator is that the estimator is independent of the used reduction technique. Therefore, it can be used for moment-matching based, Gramian matrices based or modal based model reduction techniques. The capability of the proposed technique is demonstrated by the a-posteriori error estimation of a mechanical system, and a sensitivity analysis of the parameters involved in the error estimation process is conducted.

The 3D Incompatible Element Modeling of Flexible Body in Multibody System Dynamics

An efficient 3D incompatible element for the simulation of Flexible Multibody System (FMBS) is derived based on the principle of virtual work and the theory of continuum mechanics. The presented incompatible element uses absolute coordinates and internal parameters for the degrees of freedom and fixed the locking problem under the state of pure bending. Considering the geometric nonlinearities and using the Total Lagrangian Formulation, system dynamic equations are derived and then solved by the Backward Difference Formula (BDF) in conjunction with Newton-Raphson algorithm. Eigenvalue analysis and dynamic analysis of several numerical examples are presented to show the convergence of the 3D incompatible element and excellent performance in terms of accuracy and time efficiency comparing with standard linear and quadratic hexahedral elements.

Graph-Theoretic Approach for the Dynamic Simulation of Flexible Multibody Systems

This paper provides a general description of a variational graph-theoretic formulation for simulation of flexible multibody systems (FMSs) which includes a brief review of linear graph principles required to formulate this algorithm. The system is represented by a linear graph, in which nodes represent reference frames on flexible bodies, and edges represent components that connect these frames. The method is based on a simplistic topological approach which casts the dynamic equations of motion into a symmetrical format. To generate the equations of motion with elastic deformations, the flexible bodies are discretized using two types of finite elements. The first is a 2 node 3D beam element based on Mindlin kinematics with quadratic rotation. This element is used to discretize unidirectional bodies such as links of flexible systems. The second consists of a triangular thin shell element based on the discrete Kirchhoff criterion and can be used to discretize bidirectional bodies such as high-speed lightweight manipulators, large high precision deployable space structures, and micro/nano-electromechanical systems (MEMSs). Two flexible systems are analyzed to illustrate the performance of this new variational graph-theoretic formulation and its ability to generate directly a set of motion equations for FMS without additional user input.

Investigation on the choice of boundary conditions and shape functions for flexible multi-body system

The objective of this investigation is to examine the correctness and efficiency of the choice of boundary conditions when using assumed mode approach to simulate flexible multi-body systems. The displacement field due to deformation is approximated by the Rayleigh-Ritz assumed modes in floating frame of reference (FFR) formulation. The deformations obtained by the absolute nodal coordinate (ANC) formulation which are transformed by two sets of reference coordinates are introduced as a criterion to verify the accuracy of the simulation results by using the FFR formulation. The relationship between the deformations obtained from different boundary conditions is revealed. Numerical simulation examples demonstrate that the assumed modes with cantilevered-free, simply-supported and free-free boundary conditions without inclusion of rigid body modes are suitable for simulation of flexible multi-body system with large over all motion, and the same physical deformation can be obtained using those mode functions, differ only by a coordinate transformation. It is also shown that when using mode shapes with statically indeterminate boundary conditions, significant error may occur. Furthermore, the slider crank mechanism with rigid crank is accurate enough for investigating boundary condition problem of flexible multi-body system, which cost significant less simulating time. The project was supported by the National Natural Science Foundation of China (10872126) and the Research Fund of the Doctoral Program of Higher Education of China (20100073110007).

Lie group generalized-α time integration of constrained flexible multibody systems

This paper studies a Lie group extension of the generalized-α time integration method for the simulation of flexible multibody systems. The equations of motion are formulated as an index-3 differential-algebraic equation (DAE) on a Lie group, with the advantage that rotation variables can be taken into account without the need of introducing any parameterization. The proposed integrator is designed to solve this equation directly on the Lie group without index reduction. The convergence of the method for DAEs is studied in detail and global second-order accuracy is proven for all solution components, i.e. for nodal translations, rotations and Lagrange multipliers. The convergence properties are confirmed by three benchmarks of rigid and flexible systems with large rotation amplitudes. The Lie group method is compared with a more classical updated Lagrangian method which is also formulated in a Lie group setting. The remarkable simplicity of the new algorithm opens interesting perspectives for real-time applications, model-based control and optimization of multibody systems.

Monte Carlo Simulation of Flexible Multibody System with Uncertainties

Practical mechanical systems often contain several flexible components and uncertain parameters which makes it hard to predict the system behavior and performance exactly. This research presents the uncertainty analysis of flexible multibody systems with random variables. Absolute nodal coordinate formulation (ANCF), which is different from the traditional finite element method, is employed to model the flexibility here. Monte Carlo method is successfully used to simulate flexible multibody systems of index-3. The method proposed is demonstrated by an example of flexible slider-crank mechanism.

Dynamic Simulation of Flexible Multibody System with Electro-Hydraulic Drive

Dynamic Simulation of Flexible Multibody System with Electro-Hydraulic Drive

Simulation process of flexible multibody systems with non-modal model order reduction techniques

One important issue for the simulation of flexible multibody systems is the reduction of the flexible body’s degrees of freedom. In this work, nonmodal model reduction techniques for flexible multibody systems within the floating frame of reference framework are considered. While traditionally in the multibody community modal techniques in many different forms are used, here other methods from system dynamics and mathematics are in the focus. For the reduction process, finite element data and user inputs are necessary. Prior to the reduction process, the user first needs to choose boundary conditions fitting the chosen reference frame before defining the appropriate in- and outputs. In this work, four different possibilities of modeling appropriate interface points to reduce the number of inputs and outputs are presented.

Frictional contact analysis and experiment study of complex mechanism with pinned connections

Hybrid mode aerial working vehicle (HMAWV) is a complex mechanism with many pinned connections, so it is unrealistic to get a good strength analysis by analytical method. Two numerical methods, dynamic simulation of flexible multibody systems and contact element method in FEA were developed. Contact elements including friction were discussed in depth based on a simply supported beam, and the effects of contact parameters on distribution of contact stress and global stress were analyzed by response surface method. It indicated that frictional coefficient and contact stiffness could influence distributions of contact stress tremendously, but had a trivial effect on global stress; additionally these three parameters had an important effect on the convergence of nonlinear contact analysis. Subsequently the contact element method was applied to HMAWV. In order to get a more reliable result, experimental studies were carried out. Passive strain gauges were applied to test the stresses, and then the results of tests were compared with the flexible multibody method and the contact element method. It was proved that the contact element method was more precise than the flexible multibody method under special cases. Moreover the relative error of the contact element method to experimental study was lower than 5%; therefore it had a good agreement with the experiment results.

A system‐level model reduction technique for the efficient simulation of flexible multibody systems

AbstractFlexible multibody systems are governed by sets of non‐linear differential‐algebraic equations (DAE). The number of degrees of freedom (DOFs) required for accurate body flexibility modeling, as well as the presence of the algebraic constraint equations limit the simulation speed. Body‐level model reduction and general‐purpose system‐level model reduction techniques do not result in optimal model dimension reduction and do not necessarily transform the DAE‐set into a cheaper to solve set of ordinary differential equations, as opposed to model reduction techniques, such as Global Modal Parameterization (GMP), which are based on describing the system motion by the contribution of its dominant (configuration‐dependent) system‐level modal patterns. Their use as a simulation tool has not been thoroughly investigated. This paper investigates the influence of the modal content of the configuration‐dependent mode set on the simulation accuracy through numerical experiments. By including the dynamically excited eigenmodes and static deformation patterns for DOFs in which the system is externally loaded, the system can be accurately represented by a very limited number of DOFs. Phenomena such as mode veering and mode crossing, in which the modal content of eigenmodes rapidly varies, can be expected to be problematic for such a description, which is confirmed with numerical experiments. Copyright © 2010 John Wiley & Sons, Ltd.

Forward dynamical analysis of flexible multibody systems using system-level model reduction

Fast simulation (e.g., real-time) of flexible multibody systems is typically restricted by the presence of both differential and algebraic equations in the model equations, and the number of degrees of freedom required to accurately model flexibility. Model reduction techniques can alleviate the problem, although the classically used body-level model reduction and general-purpose system-level techniques do not eliminate the algebraic equations and do not necessarily result in optimal dimension reduction. 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 the classically used fixed vector space, requiring significantly less degrees of freedom to represent the system dynamics with the same level of accuracy. The numerical experiment in this paper illustrates previously unexposed sources of approximation error: (1) the rigid body motion is computed in a forward dynamical analysis resulting in a small divergence of the rigid body motion, and (2) the errors resulting from the transformation from the modal degrees of freedom of the reduced model back to the original degrees of freedom. The effect of the configuration space discretization coarseness on the different approximation error sources is investigated. The trade-offs to be defined by the user to control these approximation errors are explained.

Erratum to: New efficient method for dynamic modeling and simulation of flexible multibody systems moving in plane

Efficient, precise dynamic analysis for general flexible multibody systems has become a research focus in the field of flexible multibody dynamics. In this paper, the finite element method and component mode synthesis are introduced to describe the deformations of the flexible components, and the dynamic equations of flexible bodies moving in plane are deduced. By combining the discrete time transfer matrix method of multibody system with these dynamic equations of flexible component, the transfer equations and transfer matrices of flexible bodies moving in plane are developed. Finally, a high-efficient dynamic modeling method and its algorithm are presented for high-speed computation of general flexible multibody dynamics. Compared with the ordinary dynamics methods, the proposed method combines the strengths of the transfer matrix method and finite element method. It does not need the global dynamic equations of system and has the low order of system matrix and high computational efficiency. This method can be applied to solve the dynamics problems of flexible multibody systems containing irregularly shaped flexible components. It has advantages for dynamic design of complex flexible multibody systems. Formulations as well as a numerical example of a multi-rigid-flexible-body system containing irregularly shaped flexible components are given to validate the method.