Analysis Of Flexible Multibody Systems Research Articles

A novel and efficient Hamiltonian dynamic analysis approach for constraint force determination in flexible multibody systems

In the dynamic analysis of flexible multibody systems, Hamiltonian formulations offer advantages in numerical stabilization and systematic handling of systems with varying mass. However, current approaches face challenges. Differential algebraic equations (DAEs) can directly express constraint forces but are computationally inefficient, while ordinary differential equations (ODEs) are more computationally efficient but cannot directly represent constraint forces due to the elimination of the Lagrange multiplier. This paper presents an innovative and efficient dynamic analysis approach based on the Hamiltonian formulation, incorporating velocity transformation and open-constraint coordinate methods. Compared to conventional DAE-based Hamiltonian formulations, our approach conducts analysis efficiently using only independent variables. Compared to conventional ODE-based Hamiltonian formulations, our approach effectively expresses constraint forces through Lagrange multiplier reformulation. Furthermore, when compared to the traditional ODE-based Lagrangian formulation, our approach exhibits superior computational efficiency. Numerical simulations assess our proposed formulation, showing agreement with conventional formulations, shorter calculation time, and alignment with analytical results, confirming the accuracy and usefulness of our approach.

Improved Craig–Bampton Method Implemented into Durability Analysis of Flexible Multibody Systems

The Craig–Bampton method is frequently applied in most commercial multibody dynamic software. Nevertheless, the Craig–Bampton modes only represent the free-free modes in flexible multibody systems. However, the free-free modes are incapable of all engineering applications. Hence, a rational set of reference conditions must be correctly chosen to define a unique displacement field. Firstly, a simple 2D beam with two revolute joints is taken as an example to prove that the free-free modes are not suitable for all engineering applications, and the results are validated by ANSYS and the analytical solution. Secondly, the Craig–Bampton method is improved by two different methods: (i) the reference conditions are added to the original Craig–Bampton matrix and (ii) the reference conditions are applied to the shape functions to redefine the mass and stiffness matrices before constructing the original Craig–Bampton matrix. This implementation illustrates that the improved Craig–Bampton matrix can not only generate the free-free modes but is also suitable for the non-free-free modes. Finally, two discrepant reference conditions are imposed to obtain the dynamic response of the flexible connecting rod based on the improved Craig–Bampton method, which is validated using the normal mode approach. Simulations show that the improved Craig–Bampton method can be used as a general-purpose method in durability analysis.

Transfer matrix method for linear vibration analysis of flexible multibody systems

Achieving high computational efficiency has been well recognized as a research challenge in the vibration analysis of flexible multibody systems. This paper presents a novel transfer matrix method to model and analyze the linear vibration of flexible multibody systems. The transfer equations, the transfer matrices, and the consistency equations of the general flexible body elements with multi-input ends are deduced for the first time. Subsequently, the automatic deduction theorem of the overall transfer equation is developed for flexible multibody systems. Based on the theorem, the overall transfer equation can be deducted automatically and the natural vibration characteristics can be obtained. The dynamic equations and augmented eigenvectors are formulated to solve the forced vibration response. Further, the proposed method is applied to study the dynamics of an ultra-precision machine tool with flexible body elements. The natural vibration characteristics and the forced vibration response are solved and validated with the data from the modal tests and the working conditions. The proposed method has the following advantages: easy deduction of the overall transfer equation, low order of system matrix, and high computational speed.

L1-norm based dynamic analysis of flexible multibody system modeled with trimmed isogeometry

Isogeometric analysis is an efficient method to accurately represent geometry and obtain accurate solutions, and can be utilized for dynamic analysis of flexible multibody systems. In this paper, l1 norm based dynamic analysis of flexible multibody system modeled with trimmed isogeometry is presented, in which, (1) the exact shape deformation of regular NURBS element after trimming is characterized by geometric transformation, (2) the l1 norm optimization is employed to select the predominant modal shapes for accurate dynamic analysis under rapidly changed loads, and (3) motion equations are sampled by considering the acceleration of control variables influence on control variables variation to obtain stable dynamic results. Finally, the presented approach is validated with three examples. The dynamic response results show that the trimmed isogeometric analysis can be employed to achieve considerable accuracy. As the continuity between NURBS elements increases from C0 to C1, the degree of freedom is reduced by 75%, and the computational time of l1 norm based dynamic analysis is about 35% of the full order modal shapes method. Furthermore, the proposed sampling method can be utilized to reduce the 33% sampling number compared with the Latin Hypercube sampling method.

The Motion Formalism for Flexible Multibody Systems

Abstract This paper describes a finite element approach to the analysis of flexible multibody systems. It is based on the motion formalism that (1) uses configuration and motion to describe the kinematics of flexible multibody systems, (2) recognizes that these are members of the special Euclidean group thereby coupling their displacement and rotation components, and (3) resolves all tensors components in local frames. The goal of this review paper is not to provide an in-depth derivation of all the elements found in typical multibody codes but rather to demonstrate how the motion formalism (1) provides a theoretical framework that unifies the formulation of all structural elements, (2) leads to governing equations of motion that are objective, intrinsic, and present a reduced order of nonlinearity, (3) improves the efficiency of the solution process, and (4) prevents the occurrence of singularities.

Formulation of Shell Elements Based on the Motion Formalism

This paper presents a finite element implementation of plates and shells for the analysis of flexible multibody systems. The developments are set within the framework of the motion formalism that (1) uses configuration and motion to describe the kinematics of flexible multibody systems, (2) couples their displacement and rotation components by recognizing that configuration and motion are members of the Special Euclidean group, and (3) resolves all tensors components in local frames. The formulation based on the motion formalism (1) provides a theoretical framework that streamlines the formulation of shell elements, (2) leads to governing equations of motion that are objective, intrinsic, and present a reduced order of nonlinearity, (3) improves the efficiency of the solution process, (4) circumvents the shear locking phenomenon that plagues shell formulations based on classical kinematic descriptions, and (5) prevents the occurrence of singularities in the treatment of finite rotation. Numerical examples are presented to illustrate the advantageous features of the proposed formulation.

Nonlinear compressed sensing-based adaptive modal shapes selection approach for efficient dynamic response analysis of flexible multibody system

Nonlinear compressed sensing-based adaptive modal shapes selection approach for efficient dynamic response analysis of flexible multibody system

Numerical investigation of the seismic response of a polar crane based on linear complementarity formulation

This paper numerically analyzes the seismic response of a polar crane operating at a nuclear plant, with derailment considered. In this case, the wheels and rails can no longer be treated as being bonded, and thus traditional methods in structural dynamics (e.g. response spectrum method) are no longer suitable for transient contact analysis. On the other hand, contact-impact analysis of flexible multibody systems will often face great difficulties attributed to the complexity of modeling and low computational efficiency. To overcome these problems, the wheel-rail interactions are modeled as multiple frictional unilateral constraints, and the resulting contact-impact problem is formulated as a linear complementarity problem (LCP). Later, a novel smoothing method is applied to smooth the LCP formulation by replacing the normal gap with the time-averaged normal gap over a short time period. A model smoothing method is then combined with modal synthesis to derive the equations of motion without high frequency components. Finally, the validity and feasibility of the proposed method are demonstrated through several numerical examples. The results show the different contact states during the seismic process, including smooth contact, detachment and re-contact.

Flexible-Link Multibody System Eigenvalue Analysis Parameterized with Respect to Rigid-Body Motion

The dynamics of flexible multibody systems (FMBSs) is governed by ordinary differential equations or differential-algebraic equations, depending on the modeling approach chosen. In both the cases, the resulting models are highly nonlinear. Thus, they are not directly suitable for the application of the modal analysis and the development of modal models, which are very useful for several advanced engineering techniques (e.g., motion planning, control, and stability analysis of flexible multibody systems). To define and solve an eigenvalue problem for FMBSs, the system dynamics has to be linearized about a selected configuration. However, as modal parameters vary nonlinearly with the system configuration, they should be recomputed for each change of the operating point. This procedure is computationally demanding. Additionally, it does not provide any numerical or analytical correlation between the eigenpairs computed in the different operating points. This paper discusses a parametric modal analysis approach for FMBSs, which allows to derive an analytical polynomial expression for the eigenpairs as function of the system configuration, by solving a single eigenvalue problem and using only matrix operations. The availability of a similar modal model, which explicitly depends on the system configuration, can be very helpful for, e.g., model-based motion planning and control strategies towards to zero residual vibration employing the system modal characteristics. Moreover, it allows for an easy sensitivity analysis of modal characteristics to parameter uncertainties. After the theoretical development, the method is applied and validated on a flexible multibody system, specifically using the Equivalent Rigid Link System dynamic formulation. Finally, numerical results are presented and discussed.

Adjoint sensitivity analysis of flexible multibody systems in differential-algebraic form

In the analysis and optimization of flexible multibody systems, an efficient sensitivity analysis can be performed using the adjoint variable method. Thereby, the procedure strongly depends on the equations of motion, which can be formulated either by minimal or by redundant coordinates yielding in turn a system of ordinary or differential-algebraic equations. In the current work, redundant coordinates are used to set an adjoint system for the sensitivity analysis of flexible multibody systems with kinematic loops, which are modeled using the floating frame of reference formulation. The procedure is tested by computing the structural gradient for the flexible crank of a slider-crank mechanism and compared with a reference procedure, where the adjoint system is formulated using minimal coordinates.

A high-precision co-rotational formulation of 3D beam elements for dynamic analysis of flexible multibody systems

Evaluating inertia forces is a central and complicated task for dynamic analysis of flexible multibody systems (FMS) involving large displacements and rotations. In a number of co-rotational approaches, cubic interpolations are adopted to formulate both inertia and internal forces, where the inertia term may be complicated in derivation and several Gauss points are needed in the numerical integration. The paper presents a high-precision co-rotational formulation of 3D beam elements for dynamic analysis of FMS. The method of switching the rotation vector and its complement is adopted to avoid the singularity problem in spatial finite rotations. In contrast to the traditional co-rotational formulation, the governing equations in the paper are formulated based on the principle of virtual power, without requiring the variation of rotation matrix. In the framework of the proposed co-rotational formulation, the stiffness matrix of small-deformation beam element can be used directly, including the Euler–Bernoulli and Timoshenko–Reissner beam elements. More essentially, the inertia terms of the beam element are formulated by discretizing the beam element into three lumped masses at the left and right end nodes, and an auxiliary node at the middle point of the beam element, resulting in a lumped mass matrix. The equivalence ensures that the total mass is completely accurate and the moment of inertia is high-precision if the beam element undergoes small elastic deformation in the local coordinate system. In the local coordinate system of co-rotational formulations, small elastic deformation can be guaranteed so that the inertia forces are formulated analytically without needing the Gauss integration. Finally, four numerical examples are considered to evaluate the accuracy of the formulation against to the geometrically exact beam theory and the previous co-rotational beam formulations. The proposed method can give high-precision numerical results from the comparison, even if using a small number of elements.

Model smoothing method of contact-impact dynamics in flexible multibody systems

An original model smoothing method for contact-impact dynamics in flexible multibody system is proposed in this work. First, the model smoothing method is applied to improve the computational efficiency, in which the instant stresses of flexible bodies are replaced by the time-averaged stresses in a short time during the modeling stage, the resulting equations of motion do not contain high frequency components. In the following, the fundamental issues of continuous contact force models are discussed. A smoothing method of modeling linear complementarity problem (LCP) is proposed for the dynamic analysis of flexible multibody system. This approach takes into account the permanent contact and impact, which has the great merit that can be used straightforward without switching contact models. Numerical results show that the model smoothing method is a new effective approach for the numerical analysis of contact-impact problems in flexible multibody system.

On the modal analysis of flexible multibody systems with singular mass and stiffness matrices

AbstractA complex mechanical system can be described as a combination of individual rigid and/or flexible components and a set of kinematical constraints. The restrictions can be either internal, which come on the scene in the case of parameterizing the configuration manifold of a system with a number of parameters larger than its intrinsic dimension, e.g. by employing three directors to describe the Lie Group SO(3) (basically for rigid bodies and beams) or by employing a single unit director to describe the two dimensional sphere with unit radius embedded in the three‐dimensional Euclidean space, i.e. S2 1 , (basically for inextensible shells, in which the component of the Green‐Lagrange strain tensor E33 is set to be zero at every time), or external ones, which come on the scene in the case of constraining bodies by means of joints, connections or supports. The linearized form of the resulting governing equations possesses usually singular mass and stiffness matrices. This fact represents an issue when the calculation of natural frequencies and natural modes for the whole system is needed. The classical modal analysis (solving directly the generalized eigenvalue problem from the stiffness and mass matrices) is no longer applicable. To overcome this difficulty, it is firstly necessary to utilize the linearized constraint equations that complete the missing information at the level of the mass and stiffness matrices and secondly, to redefine the eigenvalue and ‐vector problem within a suitable framework. In the current work, we set the focus on the development framework for the modal analysis able to deal with singular matrices.

Transient probabilistic design of flexible multibody system using a dynamic fuzzy neural network method with distributed collaborative strategy

To improve the efficiency and accuracy of transient probabilistic analysis of flexible multibody systems, a dynamic fuzzy neural network method-based distributed collaborative strategy is proposed by integrating extremum response surface method and fuzzy neural network. Distributed collaborative dynamic fuzzy neural network method is mathematically modeled and derived by considering the high nonlinearity, strong coupling, and multicomponent characteristics of a flexible multibody system. The proposed method is demonstrated to perform the transient probabilistic analysis of a two-link flexible robot manipulator. We obtain the distributional characteristics, reliability degree, and sensitivity degree of robot manipulator, which are useful for the effective design of robot manipulator. By comparing the full-scale method, extremum response surface method, dynamic fuzzy neural network method, and distributed collaborative dynamic fuzzy neural network method, we find that the distributed collaborative dynamic fuzzy neural network method can be used to perform the transient probabilistic analysis of the robot manipulator and improve the computational efficiency while maintaining a good accuracy. Moreover, this study offers a useful insight for the reliability-based design optimization of flexible multibody systems, and enriches the field of mechanical reliability theory as well.

Flexible multibody dynamics using coordinate reduction improved by dynamic correction

This paper is concerned with the efficient dynamic analysis of flexible multibody systems using a robust coordinate reduction technique. Unlike conventional static correction, the formulation is derived by dynamic correction that considers the inertia effect. In this formulation, the constraint and fixed-interface normal modes, which are representative modes in the typical coordinate reduction, are corrected by considering the truncated modal effect with the residual flexibility. Therefore, the proposed method can offer a more precise reduced system without increasing the dimension, which consequently leads to a more accurate and efficient flexible multibody simulation. We implement here the proposed method under augmented formulations of the floating reference frame approach, and test its performance with numerical examples.

Dynamics of flexible multibody systems with hybrid uncertain parameters

The most popular way to study the dynamics of a multibody system with random parameters is the probabilistic method under the assumption that the mean values and variances of the random parameters are deterministic. For many engineering systems, however, the mean values and variances of those random parameters are inherently interval parameters. In this work, a new computation method is proposed to study the dynamics of flexible multibody systems with hybrid uncertain parameters, i.e., the random parameters with interval mean values and interval variances. The flexible multibody system of concern is described by using the Absolute Nodal Coordinate Formulation first. Then, the perturbation-based method is developed in order to incorporate hybrid uncertain parameters in the dynamic analysis of flexible multibody systems. Afterwards, the dynamics equations are expanded about the mean values of the uncertain input variables by means of the second-order Taylor series. The Chebyshev collocation method, together with the scanning method, is utilized to generate the interval bounds for the deduced surrogate model. Finally, the interval mean values and interval variances of the system dynamics are obtained by using the interval arithmetic. As a consequence, the proposed method can be degenerated into the pure interval method.

A stochastic surrogate model for time-variant reliability analysis of flexible multibody system

The dynamic model of the flexible multibody systems (FMS) is usually the differential equations with time-variant, high nonlinear and strong coupling characteristics. The traditional reliability models are inefficient to solve these problems. And the reliability model is poor in accuracy and computational efficiency. Based on this point, a new stochastic surrogate model for time-variant reliability analysis of FMS is proposed. Combined model order reduction with generalized polynomial chaos, the stochastic surrogate model is established and the statistical characteristics of system responses are obtained. The calculation method of kinematic time-variant reliability is given. Finally, the effectiveness of the method is verified by a rotating flexible beam. The results show that this method has high computational accuracy compared with Monte Carlo method.

Structural sensitivity analysis of flexible multibody systems modeled with the floating frame of reference approach using the adjoint variable method

For the efficient analysis and optimization of flexible multibody systems, gradient information is often required. Next to simple and easy-to-implement finite difference approaches, analytical methods, such as the adjoint variable method, have been developed and are now well established for the sensitivity analysis in multibody dynamics. They allow the computation of exact gradients and require normally less computational effort for large-scale problems. In the current work, we apply the adjoint variable method to flexible multibody systems with kinematic loops, which are modeled using the floating frame of reference formulation. Thereby, in order to solve ordinary differential equations only, the equations of motion are brought into minimal form using coordinate partitioning, and the constraint equations at position and velocity level are incorporated in the adjoint dynamics. For testing and illustrative purposes, the procedure is applied to compute the structural gradient for a flexible piston rod of a slider–crank mechanism.

An efficient formulation based on the Lagrangian method for contact–impact analysis of flexible multi-body system

In this paper, an efficient formulation based on the Lagrangian method is presented to investigate the contact–impact problems of flexible multi-body systems. Generally, the penalty method and the Hertz contact law are the most commonly used methods in engineering applications. However, these methods are highly dependent on various non-physical parameters, which have great effects on the simulation results. Moreover, a tremendous number of degrees of freedom in the contact–impact problems will influence the numerical efficiency significantly. With the consideration of these two problems, a formulation combining the component mode synthesis method and the Lagrangian method is presented to investigate the contact–impact problems in flexible multi-body system numerically. Meanwhile, the finite element meshing laws of the contact bodies will be studied preliminarily. A numerical example with experimental verification will certify the reliability of the presented formulation in contact–impact analysis. Furthermore, a series of numerical investigations explain how great the influence of the finite element meshing has on the simulation results. Finally the limitations of the element size in different regions are summarized to satisfy both the accuracy and efficiency. In this paper, an efficient formulation based on the Lagrangian method is presented to investigate the contact–impact problems of flexible multi-body systems. A transformation from finite element coordinates p to component mode synthesis generalized coordinates y can be performed. When a contact pair is activated, a set of constraint equations should be added to the system

A hybrid finite element formulation for flexible multibody dynamics

This work deals with the transient analysis of flexible multibody systems within a hybrid finite element framework. Hybrid finite elements are based on a two-field variational formulation in which the displacements and stresses are interpolated separately yielding very good coarse mesh accuracy. Most of the literature on flexible multibody systems uses beam-theory-based formulations. In contrast, the use of hybrid finite elements uses continuum-based elements, thus avoiding the problems associated with rotational degrees of freedom. In particular, any given three-dimensional constitutive relations can be directly used within the framework of this formulation. Since the coarse mesh accuracy as compared to a conventional displacement-based formulation is very high, the scheme is cost effective as well. A general formulation is developed for the constrained motion of a given point on a line manifold, using a total Lagrangian method. The multipoint constraint equations are implemented using Lagrange multipliers. Various kinds of joints such as cylindrical, prismatic, and screw joints are implemented within this general framework. Hinge joints such as spherical, universal, and revolute joints are obtained simply by using shared nodes between the bodies. In addition to joints, the formulation and implementation details for a DC motor actuator and for prescribed relative rotation are also presented. Several example problems illustrate the efficacy of the developed formulation.