Flexible Multibody Systems Research Articles

A Novel Application of Computational Contact Tools on Nonlinear Finite Element Analysis to Predict Ground-Borne Vibrations Generated by Trains in Ballasted Tracks

Predictive numerical models in the study of ground-borne vibrations generated by railway systems have traditionally relied on the subsystem partition approach (segmented). In such a method, loads are individually applied, and the cumulative effect of the rolling stock is obtained through superposition. While this method serves to mitigate computational costs, it may not fully capture the complex interactions involved in ground-borne vibrations—especially in the frequency domain. Recent advancements in computation and software have enabled the development of more sophisticated vibrational contamination prediction models that encompass the entire dynamics of the system, from the rolling stock to the terrain, allowing continuous simulations with a defined time step. Furthermore, the incorporation of computational contact mechanics tools between various elements not only ensures accuracy in the time domain but also extends the analysis into the frequency domain. In this novel approach, the segmented models are shifted to continuous simulations where the nonlinear problem of a rigid–flexible multibody system is fully considered. The model can predict the impact of a high-speed rail (HSR) vehicle passing, capturing the key intricacies of ground-borne vibrations and their impact on the surrounding environment due to a deeper comprehension of the occurrences in the frequency domain.

Surrogate model-based cognitive digital twin for smart remote maintenance of fusion reactor: modeling and implementation

Abstract A remote maintenance robot system (RMRS) plays a critical role in safeguarding the fusion energy experimental device’s security and stability. State-of-the-art intelligent technology such as cognitive digital twins (CDT) is widely considered capable of improving complex equipment’s performance and reducing management burden using a visualized system. However, the CDT virtual space cannot mirror the RMRS which is a kind of flexible multi-body system in real-time and with high fidelity. Therefore, we propose a cognitive digital twin (CDT) modeling method based on a surrogate model for the RMRS. Firstly, model-based system engineering (MBSE) is leveraged to build a structural modular architecture, which can decrease the modeling complexity of CDT and increase the modeling efficiency. Then, the surrogate models are self-learning within the CDT physical space, which reconstructs the RMRS’s real-time dynamic performances and endows CDT with cognitive capabilities. Finally, after integrating the CDT system, a smart decision-making plan that compensates for the operation error is generated for RMRS’s accurate control. We take a China Fusion Engineering Test Reactor (CFETR) multi-purpose overload robot (CMOR) as an example to demonstrate the implementation process. According to the results, CDT can achieve real-time (230 ms time delay) high-fidelity (5 mm control error) monitoring and accurate control, and CMOR conducts smart maintenance based on the simulation results. This method improves the efficiency of RM and provides solutions for high-duty cycle time of CFETR, it can also be applied to other tokamak fusion energy devices.

Model order reduction of thermal-dynamic coupled flexible multibody system with multiple varying parameters

Spacecrafts usually suffer from the thermal shock during operation, leading to large control error or even failure. At the same time, spacecrafts are often rigid-flexible coupled multibody systems with large degrees of freedom and multiple varying parameters. The real-time control and condition monitoring require effective order reduction methods. In this investigation, the displacement and temperature fields are discretized by the Absolute Node Coordinate Formulation (ANCF) to achieve unified description. The coupled thermal-dynamic reduced-order model (ROM) is established based on the Proper Orthogonal Decomposition (POD) method. For the purpose of further increase efficiency, a linearization method is introduced so that the calculation times of the elastic force can be significantly reduced. In order to predict the response of the thermal-dynamic coupled system with multiple varying parameters, the interpolation on Grassmann manifold is introduced to maintain the orthogonality of the basis. As verification and validation, three numerical examples are presented to show the feasibility and efficacy of the proposed method.

Integrated control‐structure co‐design for flexible manipulators

AbstractThis work aims to develop a methodology for the integrated control‐structure design of flexible multibody systems. Such an issue is linked to a generalization of the inverse dynamics problem, where both the feedforward control and mechanical parameters are designed to perform a desired trajectory optimally. The resulting constrained motion is aimed at improving the dynamic performance of the system, which is underactuated, with less power demand but possible elastic deformation. Stable inversion methods are required to deal with the internal dynamics related to underactuation. The proposed methodology is based on the optimal control theory. For illustrative purposes, we considered a fully actuated linear time‐invariant (LTI) system, establishing the integrated design results in a least‐squares homogeneous system and in an optimization problem. An additional formulation is proposed for the integrated design of the flexible multibody systems, where both a multi‐criterion optimization and a scalarized problem of a corresponding nonlinear programming problem are formulated to deal with the unstable zero‐dynamics related to non‐minimum phase systems and nonlinearity. The numerical simulation of an underactuated two‐masses‐spring‐damper system and a flexible robotic manipulator in planar motions are used for validation. Finally, the beneficial aspects of the integrated design are analyzed.

Dynamic Analysis of Rigid–Flexible Structures with Piezoelectric Actuation and Control

In this paper, the dynamic equation of a rigid body coupled with two flexible beams was established, considering the rotational motion of the rigid body and the elastic deflection of the flexible beams. The coupling effect between the rigid body rotation and flexible beam vibration was studied in detail. The dynamic response of the flexible beam with piezoelectric effect was established based on Hamilton’s principle. Under the premise of ignoring the nonlinear vibration coupling term of flexible beams, only the first two modes of transverse displacement and one directional rotation motion were studied in case studies. It was found that the coupling term caused by rigid body rotation can slightly alter the natural frequency and amplitude of flexible beams, mainly reflected in the first-order mode. With increasing the external excitation on the rigid body, the induced transverse displacement at the tip of the flexible beam increased significantly. Meanwhile, when the flexible beam was under external excitation, the vibration could also induce rigid body rotation. With the inverse piezoelectric effect, one may actively actuate the flexible beam vibration and rigid body rotation or control the undesired motion of the system. The control effect can be adjusted by varying the applied voltage excitation on the piezoelectric patches as well as patch positions. The conclusions drawn from this study can be applied to active precise control of space facilities such as satellites. Based on the dynamic modeling of rigid–flexible multi-body systems, this paper completed the study of the coupling effects of rigid body rotation and flexible body vibration, and proposed an application idea for completing small-angle maneuvers of rigid–flexible multi-body systems through piezoelectric control.

Dynamic topology optimization of flexible multibody systems

Dynamic topology optimization of flexible multibody systems

Adaptive impact analysis in flexible multibody systems based on hierarchically refined IGA models

Usually, detailed impact simulation models within flexible multibody systems have to be set up manually rather than being generated automatically. This is because the process requires prior knowledge of the time and location of the impact, as well as the element resolution within the contact area. If the penalty method is used to determine the occurring contact forces, the corresponding penalty factor also needs to be determined manually. This work, however, presents an adaptive algorithm to simulate impacts within flexible multibody systems fully automatically using reduced isogeometric analysis models, the floating frame of reference formulation, and quasistatic contact models for an efficient but still accurate simulation. The adaptive algorithm detects impacts in the system, determines the contact locations on the bodies, refines the contact area, and determines the penalty factor, and therefore automatically simulates impacts. The work shows how to automatically simulate impacts in flexible multibody systems without user action or prior knowledge of impact location and size. The first application example simulates significant elastodynamic effects within a long flexible rod. The goal is to validate the algorithm by preserving the wave propagation and energy of the system. The second application example simulates the impacts of two flexible double pendulums. This setup is a suitable benchmark for the complete adaptive impact analysis procedure as the flexible double pendulums undergo large rigid body motions.

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.

Modeling and analysis for landing airbag–lunar soil interaction using a CPU–GPU-based FMBD-DEM computational framework

To investigate the landing performance of airbag cushion systems on the discrete lunar soil, this study proposes a novel modeling approach that couples multi-physics fields, including the flexible multibody systems (FMBS), the granular systems, and the gas effects. In this model, the GPU parallelism technique is utilized to address challenges arising from the large number of particles and complex contact detection involving large deformation and overall motions, and an efficient contact implementation between finite elements and particles is introduced. Additionally, contact modeling between airbags is realized using master–slave techniques. Then, a modified conventional serial staggered procedure is employed to couple flexible multibody dynamics (FMBD) and discrete element method (DEM) modules. Finally, a series of numerical examples is presented to validate the proposed dynamic model. The validated model successfully simulates the interaction between landing airbags and lunar soil. Subsequent parametric analysis contributes to the design optimization of the landing system. Moreover, discussions regarding computational efficiency illustrate that the proposed CPU–GPU-based FMBD-DEM is advantageous in engineering scenarios involving flexible components and large-scale granular systems.

Dynamic modeling of flexible multibody systems with complex geometry via finite cell method of absolute nodal coordinate formulation

Dynamic modeling of flexible multibody systems with complex geometry via finite cell method of absolute nodal coordinate formulation

Simulation on Flexible Multibody System with Topology Changes for In-space Assembly

Simulation on Flexible Multibody System with Topology Changes for In-space Assembly

A discrete adjoint gradient approach for equality and inequality constraints in dynamics

The optimization of multibody systems requires accurate and efficient methods for sensitivity analysis. The adjoint method is probably the most efficient way to analyze sensitivities, especially for optimization problems with numerous optimization variables. This paper discusses sensitivity analysis for dynamic systems in gradient-based optimization problems. A discrete adjoint gradient approach is presented to compute sensitivities of equality and inequality constraints in dynamic simulations. The constraints are combined with the dynamic system equations, and the sensitivities are computed straightforwardly by solving discrete adjoint algebraic equations. The computation of these discrete adjoint gradients can be easily adapted to deal with different time integrators. This paper demonstrates discrete adjoint gradients for two different time-integration schemes and highlights efficiency and easy applicability. The proposed approach is particularly suitable for problems involving large-scale models or high-dimensional optimization spaces, where the computational effort of computing gradients by finite differences can be enormous. Three examples are investigated to validate the proposed discrete adjoint gradient approach. The sensitivity analysis of an academic example discusses the role of discrete adjoint variables. The energy optimal control problem of a nonlinear spring pendulum is analyzed to discuss the efficiency of the proposed approach. In addition, a flexible multibody system is investigated in a combined optimal control and design optimization problem. The combined optimization provides the best possible mechanical structure regarding an optimal control problem within one optimization.

A nonsmooth modified symplectic integration scheme for frictional contact dynamics of rigid–flexible multibody systems

Constructing an efficient high-order nonsmooth time-stepping integration scheme for rigid–flexible multibody systems with frictional contacts and impacts is a challenging task. This paper presents a nonsmooth modified symplectic integration scheme that can automatically achieve second-order convergence if possible. This scheme is obtained in three main steps. Firstly, the position coordinates are used to interpolate midpoint approximated integrals, which include velocity jumps implicitly; Secondly, a two-timestep central difference scheme is formulated to approximate the measure differential inclusions; Thirdly, composing two half-condensed two-timestep schemes leads to a one-timestep scheme, which retains the structure of a second-order scheme. Furthermore, while bilateral constraints are satisfied at the position level, unilateral constraints at the velocity level are subject to constraint drifts. To alleviate this issue, a classified cone complementarity problem is formulated to restrain penetration from deteriorating and sticking contacts from fictitious slipping. After time discretization, a set of nonlinear equations with cone complementarity conditions at each timestep is solved by a two-layer iterative computational strategy, where the outer layer is Newton iteration and the inner layer is a primal–dual interior point method. Several numerical examples demonstrate the high accuracy attained by the proposed method. Nonsmooth dynamics of a rigid–flexible robot involving high-speed and high-friction impacts can be simulated faster than real-time.

Exudyn – a C++-based Python package for flexible multibody systems

The present contribution introduces the design, methods, functionalities, and capabilities of the open-source multibody dynamics code Exudyn, which has been developed since 2019. The code has been designed for rigid and flexible multibody systems, with a focus on performance for multicore desktop processors. It includes script-language-based modeling and it is intended to be used in science and education, but also in industry. The open-source code is available on GitHub and consists of a main C++ core, a rich Python interface including pre- and postprocessing modules in Python, and a collection of rigid and flexible bodies with appropriate joint, load, and sensor functionality. Integrated solvers allow explicit and implicit time integration, static solution, eigenvalue analysis, and optimization. In the paper, the code design, structure, computational core, computational objects, and multibody formulations are addressed. In addition, the computational performance is evaluated with examples of rigid and flexible multibody systems. The results show the significant impact of multithreading especially for small systems, but also for larger models.

Absolute nodal coordinate formulation – Multilevel finite element framework for the nonlinear multi-scale multibody dynamic analysis of composite structures

In this paper, a multi-scale modeling approach for the dynamic response of flexible multibody system made of composite material is suggested using the absolute nodal coordinate formulation (ANCF) – multilevel finite element (FE2) method. On many occasions, the analysis of composite structures requires a multi-scale modeling approach. FE2 method is one of the famous multi-scale modeling methods. And it is a versatile computational homogenization approach that is commonly applicable to many different materials. Because the present FE2 is capable of computing the mechanical response at two different scales simultaneously, ANCF element will be used at the macroscopic scale. And the microscopic RVE at each macroscopic integration point will be obtained by the finite element method. Since ANCF element uses only the absolute nodes and gradients, the mass matrix will be constant. Due to such characteristics, when combined with FE2, the linearization process will become easier compared against the existing formulation. And Coriolis or centrifugal force will not need to be considered in the dynamic formulation. Therefore, computational cost will be reduced, and the relevant algebraic manipulation will become simplified. Several numerical examples are presented to demonstrate the improved accuracy and diminished computational cost of the present suggestion.

A CPU parallelized coupling framework for constrained flexible multibody system and granular matter based on contact-graph

This paper proposes a coupling model for simulating the interaction between particles and the flexible multibody system. The flexible rods are described using the Absolute Nodal Coordinate Formulation, and various types of contacts, including particle-particle, particle-rod, particle-rigid body contacts, and contacts within multibody objects, are considered. The contact detection is improved by combining space partitioning techniques and the bounding box method. A data structure based on a contact graph is introduced to store the contact relationships efficiently. The discrete element system and multibody system are coupled through serial time integration, with data exchanges at the program level. Numerical examples demonstrate the validity of the proposed model. Moreover, the model is applied to simulate the asteroid soil sampling with flexible brush sampler, showing good agreement with experimental data. The analysis of engineering examples provides insights for sampler design, and show the model's engineering value.

Highly accurate differentiation of the exponential map and its tangent operator

Exponential coordinates are widely used in simulation codes for flexible multibody systems based on a Lie group approach. The accurate and efficient evaluation of the exponential map, the tangent operator and its higher order derivatives is crucial. This paper presents a systematic derivation process based on the matrix series expansion of the exponential map and of the tangent operator. This approach is general as it can be applied to any matrix Lie group. For the Lie groups SO(3) and SE(3), the closed form expression of these operators can also be established and is summarized in the paper. It is shown that the closed form of the operators is affected by round-off errors for small rotation amplitudes, whereas the series form is affected by truncation errors at high rotation amplitudes. The computational efficiency of the two approaches is also discussed. A switching strategy between the closed form and the series form is then proposed to obtain an adjustable compromise between accuracy and computational cost.

Non-smooth numerical solution for Coulomb friction, rolling and spinning resistance of spheres applied to flexible multibody system dynamics

Non-smooth numerical solution for Coulomb friction, rolling and spinning resistance of spheres applied to flexible multibody system dynamics

Strongly coupled fluid–structure interaction analysis of aquatic flapping wings based on flexible multibody dynamics and the modified unsteady vortex lattice method

Because of their superior efficiency and low detectability compared to conventional submarine-shaped vehicles, bionic underwater vehicles such as Mantabot are playing increasingly important roles in ocean exploration. However, current research on manta-inspired robots is limited to structural design and pure hydrodynamics analysis of pectoral fin performance, with other aspects such as hydroelasticity and optimal design of flapping wings rarely addressed. This work presents a novel method for analyzing aquatic flapping wings by coupling flexible multibody dynamics with a modified unsteady vortex lattice method. The flexible multibody system is solved by using the absolute nodal coordinate formulation, and the conventional unsteady vortex lattice method is modified to accommodate dynamic stall. Compactly supported radial basis functions are used to transfer the hydroelastic forces and structural displacements across the interface meshes while satisfying global energy conservation, and a predictor–corrector method is used to stabilize the iteration procedure. Three different experiments are used to validate the flexible multibody dynamics solver, the modified unsteady vortex lattice solver, and the proposed fluid–structure interaction framework, respectively. Finally, the hydroelastic framework is demonstrated for a fin-like wing, and how flexibility and movement affect the flapping wing dynamics is examined. Numerical examples show that reducing the structural stiffness of the aquatic flapping wing within a certain range improves the propulsive efficiency but also reduces the thrust coefficient, and how the stiffness distribution affects the thrust and propulsive efficiency remains consistent over a range of the Strouhal number.

A new sandwich beam model with layer-to-layer boundary modified displacements based on higher-order absolute nodal coordinate formulation

The absolute nodal coordinate formulation (ANCF) is one of the efficient modeling methods to solve the dynamics of flexible multibody systems. Zig-zag theory can accurately calculate the stress variation in the sandwich structure with only a few unknown variables, thus providing a good balance between computational efficiency and computational accuracy. Combining ANCF and zig-zag theory, this paper proposes a new sandwich beam model, called HANCF-ZZ, by introducing layer-to-layer boundary modified displacements in a high-order ANCF beam model. The capability of HANCF-ZZ for static deformation analysis and free vibration analysis is verified by comparing experimental, analytical, and numerical results in other literature. The results show that the HANCF-ZZ corrects the global stiffness overestimated by the equivalent single-layer theory, eliminates the locking problem of the low-order ANCF, and captures the nonlinear bifurcation phenomenon in large deformation problems under large loads. The HANCF-ZZ is not only an efficient numerical method for zig-zag theory but also enriches the ANCF element types to enable its application to the dynamics analysis of soft core sandwich beams.