Flexible Multibody Systems Research Articles

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.

Model order reduction based on successively local linearizations for flexible multibody dynamics

SummaryAn efficient method of model order reduction is proposed for the dynamic computation of a flexible multibody system undergoing both large overall motions and large deformations. The system is initially modeled by using the nonlinear finite elements of absolute nodal coordinate formulation and then locally linearized at a series of quasi‐static equilibrium configurations according to the given accuracy in dynamic computation. By using the Craig‐Bampton method, the reduced model is established by projecting the incremental displacements of the locally linearized system onto a set of local modal bases at the quasi‐static equilibrium configuration accordingly. Afterwards, the initial conditions for the dynamic computation for the reduced model via the generalized‐α integrator can be determined from the modal bases. The analysis of computation complexity is also performed. Hence, the proposed method gives time‐varying and dimension‐varying modal bases to elaborate the efficient model reduction. Finally, three examples are presented to validate the accuracy and efficiency of the proposed method.

Flexible Multibody System Dynamics by Means of a Spectral Based Meshless Approach

Flexible Multibody System Dynamics by Means of a Spectral Based Meshless Approach

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.

Constrained Topology Optimization of Dynamic Mechanical Systems

AbstractWe present a topology optimization procedure for flexible multibody systems, which is subjected to time‐ and space‐dependent constraints. The flexible bodies are parameterized using the solid isotropic material with penalization approach. For the efficient solution of the optimization problem, the pointwise time‐dependent constraints are discretized and aggregated into a single constraint using the Kreisselmeier‐Steinhauser function. The procedure is tested by optimizing the mass of a flexible piston rod, while the deformation within the design domain is constrained.

Gradient Approximation in Flexible Multibody Systems Using a Multigrid Approach

AbstractThe approximation of gradients with a multigrid approach is presented in the framework of flexible multibody systems, in which the flexible bodies are modeled using finite element method. For the cases where the number of design variables are high, for example, in topology optimization where the density of each element is chosen as a design variable, the gradient computation becomes computationally infeasible. The proposed approximation, in which the gradients are computed simultaneously on a fine grid and a coarse grid, can significantly reduce the computation time without hindering the accuracy of the gradients.

A novel 3D variational aeroelastic framework for flexible multibody dynamics: Application to bat-like flapping dynamics

We present a novel three-dimensional (3D) variational aeroelastic framework for flapping wing with a flexible multibody system subjected to an external incompressible turbulent flow. The proposed aeroelastic framework consists of a three-dimensional fluid solver with a hybrid RANS/LES model based on the delayed detached eddy simulation (DDES) treatment and a nonlinear monolithic elastic structural solver for the flexible multibody system with constraints. Radial basis function (RBF) is applied in this framework to transfer the aerodynamic forces and structural displacements across the discrete non-matching interface meshes while satisfying global energy conservation. For the consistency of the interface data transfer process, the mesh motion of the fluid domain with large elastic deformation due to high-amplitude flapping motion is also performed via the standard radial basis functions. The fluid equations are discretized using a stabilized Petrov–Galerkin method in space and the generalized-α approach is employed to integrate the solution in time. The flexible multibody system is solved by using geometrically exact co-rotational finite element method and an energy decaying scheme is used to achieve numerical stability of the multibody solver with constraints. A nonlinear iterative force correction (NIFC) scheme is applied in a staggered partitioned iterative manner to maintain the numerical stability of aeroelastic coupling with strong added mass effect. An isotropic aluminum wing with flapping motion is simulated via the proposed aeroelastic framework and the accuracy of the coupled solution is validated with the available experimental data. We next study the robustness and reliability of the 3D flexible multibody aeroelastic framework for an anisotropic flapping wing flight involving battens and membranes with composite material and compare against the experimental results. Finally, we demonstrate the aeroelastic framework for a bat-like wing and examine the effects of flexibility on the flapping wing dynamics.

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.

Deployment dynamical numerical simulation on large elliptical truss antenna

With the unique advantages in the application of bias-feed high-precision mesh antennas, the large elliptical truss antenna (LETA) attracts attention of many scholars. Due to the complexity of the LETA deployment process, there are some potential risks in orbit development. The successful deployment of antenna on orbit is the key to determine the success or failure of satellite missions. In order to improve the deployment reliability and the load safety of the components of the antennas, the dynamic characteristics of the LETA must be fully identified by the means of simulation, and the load characteristics of the components during the deploying process must be obtained. As a typical flexible multi-body system, a geometric precise beam is used to model bars with large deformation in the LETA and the flexible driving cable is modeled based on one-dimensional dynamic medium dynamics theory. The dynamic modeling and numerical simulation of the deploying process are presented in the paper. On this basis, the unsynchronized phenomenon is analyzed, which is caused mainly by the flexibility of components and the friction loss of driving cable through the hinges. In addition, from the perspective of energy, it shows that friction loss and mesh tension are the most important factors affecting the loads of driving cable and truss. Finally, the design optimization suggestions are given to improve the reliability.

A projection-based approach for the derivation of the floating frame of reference formulation for multibody systems

The reduction in the number of coordinates for flexible multibody systems is necessary in order to achieve acceptable simulation times of real-life structures and machines. The conventional model order reduction technique for flexible multibody systems is based on the floating frame of reference formulation (FFRF), using a rigid body frame and superimposed small flexible deformations. The FFRF leads to strongly coupled terms in rigid body and flexible coordinates as well as to a non-constant mass matrix. As an alternative to the FFRF, a formulation based on absolute coordinates has been proposed which uses a co-rotational strain. In this way, a constant mass matrix and a co-rotational stiffness matrix are obtained. In order to perform a reduction in the number of coordinates, by means of the component mode synthesis, e.g., the number of modes needs to be increased, such that all modes are represented in every possible rotated configuration. This approach leads to the method of generalized component mode synthesis (GCMS). The present paper shows in detail how the equations of motion of the FFRF evolve from the ones of the GCMS by considering rigid body constraint conditions and subsequently eliminating them via an appropriate null-space projection. This approach allows a straightforward, term-by-term interpretation of the FFRF mass matrix and of the generalized gyroscopic forces, which, to the same extent, cannot be deduced from former publications on the FFRF. From a practical point of view, the resulting expressions allow to calculate all inertia coefficients from the constant finite element mass matrix together with standard input data of the finite element model in the course of a preprocessing step. Then, the repeated updates of the FFRF mass matrix and of the gyroscopic forces in the course of time integration involve only simple vector matrix operations of low dimensions. In contrast to previous implementations of the FFRF, no evaluations of extra inertia integrals are required. Consequently, the present formulation can be implemented entirely independent of the related finite element code.

A modal derivatives enhanced Rubin substructuring method for geometrically nonlinear multibody systems

This paper presents a novel model order reduction technique for 3D flexible multibody systems featuring nonlinear elastic behavior. We adopt the mean-axis floating frame approach in combination with an enhanced Rubin substructuring technique for the construction of the reduction basis. The standard Rubin reduction basis is augmented with the modal derivatives of both free-interface vibration modes and attachment modes to consider the bending–stretching coupling effects for each flexible body. The mean-axis frame generally yields relative displacements and rotations of smaller magnitude when compared to the one obtained by the nodal-fixed floating frame. This positively impacts the accuracy of the reduction basis. Also, when equipped with modal derivatives, the Rubin method better considers the geometric nonlinearities than the Craig–Bampton method, as it comprises vibration modes and modal derivatives featuring free motion of the interface. The nonlinear coupling between free-interface modes and attachment modes is also considered. Numerical tests confirm that the proposed method is more accurate than Craig–Bampton’s, a nodal fixed floating frame counterpart originally proposed in Wu and Tiso (Multibody Syst. Dyn. 36(4): 405–425, [2016]), and produces significant speed-ups. However, the offline cost is increased because the mean-axis formulation produces operators with decreased sparsity patterns.

Dynamic contact model of shell for multibody system applications

In a multibody system consisting of shell structures, the contact may appear in any area of shells. It is difficult to simulate the contact of shells with large deformation because of the geometric nonlinearity of deformation and the boundary nonlinearity of contact. This study presents a rotation-free shell formulation and an extended contact discretization in multibody systems using a corotational frame. This model is different from previous formulations in the definition of the local frame and the processing of local large curvature. In order to deal with the shell contact, a unified contact discretization scheme including edge-to-edge contact for facet triangle shell elements is proposed to solve the large penetration problem. A series of numerical examples of multibody dynamics have validated the approach of the nonlinear shell model and contact treatments. Moreover, a practical application of deployment of solar cells shows the capability of the proposed formulation in solving large-scale problems of flexible multibody system with large deformation and contact.

A formulation for modeling levitation based vibration energy harvesters undergoing finite motion

This paper presents a geometrically exact formulation for modeling electromagnetic vibration energy harvesters undergoing finite displacements and rotations. The formulation is conceived to model uniaxial harvesters that extract energy from the vibration of flexible multibody systems that are moving arbitrarily in space. The finite rotation framework is based on an updated Lagrangian procedure; the geometrically exact dynamic equilibrium equations of the harvester are written in terms of local director vectors using a material description. The resulting coupled electro-mechanical equations of motion preserve its classical one-dimensional form. Numerical tests are conducted to address the response of the harvester under specific finite displacements-finite rotations excitations.

Simultaneous topology and size optimization of a 3D variable-length structure described by the ALE–ANCF

An efficient optimization approach is proposed to simultaneously optimize the topology and geometrical sizes of a three-dimensional (3D) variable-length structure. First of all, in a unified global coordinate system, a flexible multibody system (FMBS) model is established to describe the extension and contraction dynamics of a variable-length structure. The variable-length flexible structure is meshed by the proposed arbitrary Lagrangian–Eulerian (ALE) solid elements of absolute nodal coordinate formulation (ANCF). To consider the variable-length behaviors, the classic equivalent static loads (ESL) is redefined by introducing the concept of virtual design domain. With the help of the ESL, the dynamic response optimization of the variable-length structure is turned into a static one. A simultaneous topology and size optimization approach is initially proposed via the moving morphable components (MMC). By appropriately arranging the initial configuration of the components and linking all or part of the design variables of a component with others’, the variable-length structure can be designed as a homogeneous or heterogeneous periodic structure. The approach is numerically implemented via an additional constraint that modifies the sensitivities with respect to some design variables. Finally, three numerical examples are presented to demonstrate the effectiveness and efficiency of the proposed optimization approach.

Modeling and analysis of sliding joints with clearances in flexible multibody systems

The modeling of the sliding joint with clearance between a flexible beam and a rigid hole is investigated in this paper. The flexible beam is discretized using the three-dimensional curved Euler–Bernoulli beam element of the Absolute Nodal Coordinate Formulation, while the motion of the rigid hole is described by the Cartesian coordinates. Moreover, the cross sections of both the flexible beam and the rigid hole are assumed to be circular. The existing joints with clearances are mainly rigid joints with small clearances, and the contact detection algorithm adopted can solve only one pair of potential contact points within one section. In order to model the contact problem in the sliding joint with clearance, a new contact detection method based on the intersection of the rigid hole’s cross section and the flexible beam is proposed, which yields a two-dimensional contact detection problem. Based on the common-normal concept, the ellipse–circle contact detection problem within the hole’s cross section can be solved. The potential contact point on the hole’s cross section will be determined, and the closest point projection on the beam’s neutral axis can be defined further. The proposed contact detection method can deal with the sliding joint with large clearance and the multiple-point contact problem within one section. In addition, the penalty method is adopted to model the frictionless contact between the flexible beam and the rigid hole. Finally, two numerical examples about sliding joints with clearances, one with an initially curved beam under gravity and the other with a straight beam under zero gravity, are presented to demonstrate the influence of the clearance of sliding joint on the dynamic performance of flexible multibody systems.

Smoothed particle hydrodynamics and modal reduction for efficient fluid–structure interaction

ABSTRACTFrom wind turbines to capillary blood flow, problems of fluid–structure interaction occur across different scales of length and time. Owed to the multiple scales involved, the inherent complexity of coupled fluid and structural dynamics requires advanced numerical methods for the computational analysis. The proposed method relies on the coupling of modally reduced flexible multibody systems with fluids represented by the smoothed particle hydrodynamics method. A reduced-order basis is employed to describe small flexible deformation of a structural component relative to its large rigid body motion that is represented by a body-fixed frame. An efficient evaluation of the surface deformation of the bodies involved is a vital ingredient for the coupling. Conventional approaches suffer from the drawback of flexible deformation being represented in a body-fixed local frame. The generalized component mode synthesis, in which the total displacement is interpolated, mitigates this computational limitation. The linear relation between generalized coordinates and the total displacement allows the coupling forces to be evaluated within the parallel fluid framework. Only the reduced set of equations of motion is solved on the solid side using an implicit time integration scheme with possibly large time-steps, whereas the particle-based fluid simulation relies on a fast explicit scheme.

A flexible natural coordinates formulation (FNCF) for the efficient simulation of small‐deformation multibody systems

SummaryThis paper introduces the novel flexible natural coordinates formulation to model small‐deformation multibody dynamics. The main contribution of this work is its resulting constant mass matrix and quadratic constraint equations devoid of any other nonlinearities. These properties are similar to those of a natural coordinates formulation for rigid multibody systems with the addition of constant damping and stiffness matrices to model the flexibility under the assumption of small deformations. As such, it is a straightforward extension to natural coordinates while maintaining its beneficial properties. The main concept of the current approach is to introduce ample redundancy in the set of generalized coordinates to simplify the kinematics ensuring the aforementioned properties and the similarity to a natural coordinates approach. This is not achievable by standard techniques that introduce redundancy. Not only does this offer a very simple equation structure but also interesting properties toward the development of system‐level model order reduction techniques for flexible multibody systems as well as a straightforward parameter gradient extraction. The formulation accuracy is validated with a floating frame of reference implementation.

Monitoring of a controlled space flexible multibody by means of embedded piezoelectric sensors and cameras synergy

Interaction between elastic dynamics and attitude control is a serious problem in space operations, which often involve satellites with highly flexible appendages. Monitoring and eventually control of the vibrations are a major concern to avoid a decrease in the expected performance. In particular, the classic case of a central bus with two lateral appendages (solar panels) is considered. The design of a system for structural vibration monitoring is proposed both from a numerical and an experimental point of view. Piezoelectric devices are a usual solution for measuring the deformation of the structures. In the proposed work, optical sensors are also implemented: the combined use of the two sets allows for the monitoring of the elastic displacement of the solar panels and for the reconstruction of the modal shapes of the entire flexible multibody system.

Dynamics of flexible multibody systems with variable-speed control moment gyroscopes

This paper presents a generic global matrix formulation for the dynamics of flexible multibody systems with variable-speed control moment gyroscopes (VSCMGs). The flexible bodies are assumed to exhibit only small deformation, and they are connected in a tree topology by hinges permitting large rotation and translation. A cluster of VSCMGs is mounted on each body for actuation; it is assumed that the VSCMGs are statically and dynamically balanced, and their rotors are axisymmetric. A minimum set of dynamic equations are derived systematically via a mixed use of Kane's method and Newton–Euler equations. The parameters of each flexible body are augmented to take into account the inertias of the attached VSCMGs. Moreover, a skew-symmetric gyroscopic matrix and three control-input-mapping matrices are defined to represent the passive and the active gyroscopic torques of the VSCMGs in a global matrix manner. Three examples are given to show the usefulness, versatility, and correctness of the proposed formulation. As an additional contribution, also presented are linear dynamic equations for a spacecraft with flexible appendages and embedded VSCMGs.

A Method for Calculating and Continuing Static Solutions for Flexible Multibody Systems

A method to calculate static solutions for mechanical systems containing rigid and flexible bodies modeled by finite elements is described. The formulation of the equations makes use of generalized strains, which leads to an extended set of equations for both these generalized strains and nodal coordinates, together with constraint equations imposing the relations between these two groups of coordinates. The associated Lagrangian multipliers are the generalized stresses. The resulting iteration scheme appears to be quite robust in comparison with more traditional methods, especially if some displacements are prescribed. Once a static solution has been found, the linearized equations of motion about this solution can be obtained in terms of a set of minimal coordinates, that is, in the degrees-of-freedom (DOFs). In addition, a continuation method is described for tracing a branch of static solutions if some parameters are varied. The method is of the familiar predictor–corrector type with a linear or cubic predictor and a corrector with a step size constraint. Applications to a large-deflection problem of a curved cantilever beam, large deflections of a fluid-conveying tube and its resulting instability, and the buckling of an overconstrained parallel leaf-spring mechanism due to misalignment are given.