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

Multibody System Dynamics - 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...     

An efficient formulation for flexible multibody dynamics using a condensation of deformation coordinates

Multibody System Dynamics - In this work a new approach to deal with non-ideal operative aspects of spatial revolute joints by means of a three-dimensional finite element analysis (3D-FEA) is...     

Impulse-based substructuring in a floating frame to simulate high frequency dynamics in flexible multibody dynamics

An original approach for flexible multibody dynamics is proposed, which combines the free–free formulation of elastic body deformation with an impulse-based representation of linear vibration. The resulting system of equations being remarkably simple, this impulse-based substructuring method is straightforward to implement. Simple applications of a flexible rotating beam submitted to various excitation inputs have been selected and developed so as to assess the accuracy of the proposed methodology.     

Moving loads on flexible structures presented in the floating frame of reference formulation

The introduction of moving loads in the Floating Frame of Reference Formulation is presented. We derive the kinematics and governing equations of motion of a general flexible multibody system and their extension to moving loads. The equivalence of convective effects with Coriolis and centripetal forces is shown. These effects are measured numerically and their significance in moving loads traveling at high speed is confirmed. A method is presented to handle discontinuities when moving loads separate from the flexible structure. The method is extended from beam models to general flexible structures obtained by means of the Finite Element Method. An interpolation method for the deformation field of the modal representation of these bodies is introduced.The work is concluded by application of the method to modern mechanical problems in numerical simulations.     

The development of a sliding joint for very flexible multibody dynamics using absolute nodal coordinate formulation

In this paper, a formulation for a spatial sliding joint is derived using absolute nodal coordinates and non-generalized coordinate and it allows a general multibody move along a very flexible cable. The large deformable motion of a spatial cable is presented using absolute nodal coordinate formulation, which is based on the finite element procedures and the general continuum mechanics theory to represent the elastic forces. And the nongeneralized coordinate, which is related to neither the inertia forces nor the external forces, is used to describe an arbitrary position along the centerline of a very flexible cable. Hereby, the non-generalized coordinate represents the arc-length parameter. The constraint equations for the sliding joint are expressed in terms of generalized coordinate and nongeneralized coordinate. In the constraint equations for the sliding joint, one constraint equation can be systematically eliminated. There are two independent Lagrange multipliers in the final system equations of motion associated with the sliding joint. The development of this sliding joint is important to analyze many mechanical systems such as pulley systems and pantograph-catenary systems for high speed-trains.     

On the influence of model reduction techniques in topology optimization of flexible multibody systems using the floating frame of reference approach

Employing the floating frame of reference formulation in the topology optimization of dynamically loaded components of flexible multibody systems seems to be a natural choice. In this formulation the deformation of flexible bodies is approximated by global shape functions, which are commonly obtained from finite element models using model reduction techniques. For topology optimization these finite element models can be parameterized using the solid isotropic material with penalization (SIMP) approach. However, little is known about the interplay of model reduction and SIMP parameterization. Also securing the model reduction quality despite major changes of the design during the optimization has not been addressed yet. Thus, using the examples of a flexible frame and a slider-crank mechanism this work discusses the proper choice of the model reduction technique in the topology optimization of flexible multibody systems.     

Reanalysis-based fast solution algorithm for flexible multi-body system dynamic analysis with floating frame of reference formulation

Multibody System Dynamics - In order to improve the computational efficiency of flexible multi-body system dynamic analysis with floating frame of reference formulation (FFRF), a reanalysis-based...     

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.     

First order sensitivity analysis of flexible multibody systems using absolute nodal coordinate formulation

Design sensitivity analysis of flexible multibody systems is important in optimizing the performance of mechanical systems. The choice of coordinates to describe the motion of multibody systems has a great influence on the efficiency and accuracy of both the dynamic and sensitivity analysis. In the flexible multibody system dynamics, both the floating frame of reference formulation (FFRF) and absolute nodal coordinate formulation (ANCF) are frequently utilized to describe flexibility, however, only the former has been used in design sensitivity analysis. In this article, ANCF, which has been recently developed and focuses on modeling of beams and plates in large deformation problems, is extended into design sensitivity analysis of flexible multibody systems. The Motion equations of a constrained flexible multibody system are expressed as a set of index-3 differential algebraic equations (DAEs), in which the element elastic forces are defined using nonlinear strain-displacement relations. Both the direct differentiation method and adjoint variable method are performed to do sensitivity analysis and the related dynamic and sensitivity equations are integrated with HHT-I3 algorithm. In this paper, a new method to deduce system sensitivity equations is proposed. With this approach, the system sensitivity equations are constructed by assembling the element sensitivity equations with the help of invariant matrices, which results in the advantage that the complex symbolic differentiation of the dynamic equations is avoided when the flexible multibody system model is changed. Besides that, the dynamic and sensitivity equations formed with the proposed method can be efficiently integrated using HHT-I3 method, which makes the efficiency of the direct differentiation method comparable to that of the adjoint variable method when the number of design variables is not extremely large. All these improvements greatly enhance the application value of the direct differentiation method in the engineering optimization of the ANCF-based flexible multibody systems.     

On the use of moment-matching to build reduced order models in flexible multibody dynamics

An important issue in the field of flexible multibody dynamics is the reduction of the flexible body's degrees of freedom. For this purpose, often modal reduction through projection onto a subspace spanned by some dominant eigenvectors is used. However, as in this method the dynamical boundary conditions are not taken into account, a large number of eigenmodes is required to obtain a good approximation and also the selection of the dominant modes can be quite difficult. Therefore, the authors propose an approach based on accounting for the flexible body as an input-output system in the frequency domain. The reduced order model is generated by imposing a set of interpolation conditions concerning the values and derivatives of the system's transfer function in a predefined frequency range. This procedure is known as moment-matching and can be realised through projection onto so-called Krylov-subspaces. As this technique allows the incorporation of the frequency content and the spatial distribution of the loads, in the chosen frequency range more accurate reduced order models can be obtained compared to other model reduction techniques available in structural mechanics. The calculation of the Krylov-subspaces can be implemented very efficiently, using the Arnoldi or Lanczos procedure in connection with sparse matrix techniques. The capability of the proposed technique is demonstrated by means of a numerical example.     

Shock response analysis of hard disk drive using flexible multibody dynamics formulation

Recent development of the shock analysis on the HDD is briefly reviewed. A flexible multi-body dynamics formulation is developed to simulate the shock response of the HDD. If one component in the HDD is changed, only mode shapes and frequencies of that component should be re-calculated and then used to obtain the system’s response. Steady state Reynolds equation is solved to obtain the air pressure on the slider and disk for various slider positions. An air pressure table is formed and used to model the non-linear air bearing during the simulation. Responses of flying height for different direction and shock duration time are analyzed. Results show that the flying state of the slider is more sensitive to the shock with shorter duration time.     

Static form-finding analysis of a railway catenary using a dynamic equilibrium method based on flexible multibody system formulation with absolute nodal coordinates and controls

This paper proposes a dynamic equilibrium method for finding the initial equilibrium configuration of a railway catenary. In the proposed method, the catenary composed of flexible wires is modeled using two-node cable elements with absolute nodal coordinates based on a flexible multibody system formulation. Dynamic conditions that characterize the initial equilibrium configuration of the catenary are given and addressed as control processes in the form-finding procedure. The key feature of the proposed method is that the catenary configuration is continually evolved by dynamic simulation until characterization conditions are attained and an equivalent configuration of the centenary at static equilibrium is thus computed. It is validated using two examples and applied to the form-finding analysis of a two dimensional simple railway catenary. The obtained results are analyzed and discussed. It is general and can be applied to catenaries with complex configurations.     

Isogeometric shell discretizations for flexible multibody dynamics

This work aims at including nonlinear elastic shell models in a multibody framework. We focus our attention to Kirchhoff–Love shells and explore the benefits of an isogeometric approach, the latest development in finite element methods, within a multibody system. Isogeometric analysis extends isoparameteric finite elements to more general functions such as B-splines and NURBS (Non-Uniform Rational B-Splines) and works on exact geometry representations even at the coarsest level of discretizations. Using NURBS as basis functions, high regularity requirements of the shell model, which are difficult to achieve with standard finite elements, are easily fulfilled. A particular advantage is the promise of simplifying the mesh generation step, and mesh refinement is easily performed by eliminating the need for communication with the geometry representation in a CAD (Computer-Aided Design) tool. Target applications are wind turbine blades and twist beam rear suspensions. First numerical examples demonstrate an impressive convergence behavior of the isogeometric approach even for a coarse mesh, while offering substantial savings with respect to the number of degrees of freedom.     

A concise nodal-based derivation of the floating frame of reference formulation for displacement-based solid finite elements

Multibody System Dynamics - The Floating Frame of Reference Formulation (FFRF) is one of the most widely used methods to analyze flexible multibody systems subjected to large rigid-body motion but...     

Non-linear dynamics of flexible multibody systems

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

Linear dynamics of flexible multibody systems

We consider mechanical systems where the dynamics are partially constrained to prescribed trajectories. An example for such a system is a building crane with a load and the requirement that the load moves on a certain path.Enforcing this condition directly in form of a servo constraint leads to differential-algebraic equations (DAEs) of arbitrarily high index. Typically, the model equations are of index 5, which already poses high regularity conditions. If we relax the servo constraints and consider the system from an optimal control point of view, the strong regularity conditions vanish, and the solution can be obtained by standard techniques.By means of the well-known \(n\)-car example and an overhead crane, the theoretical and expected numerical difficulties of the direct DAE and the alternative modeling approach are illustrated. We show how the formulation of the problem in an optimal control context works and address the solvability of the optimal control system. We discuss that the problematic DAE behavior is still inherent in the optimal control system and show how its evidences depend on the regularization parameters of the optimization.     

A two-step approach for model reduction in flexible multibody dynamics

In this work, a two-step approach for model reduction in flexible multibody dynamics is proposed. This technique is a combination of the Krylov-subspace method and a Gramian matrix based reduction approach that is particularly suited if a small reduced-order model of a system charged with many force-inputs has to be generated. The proposed methodology can be implemented efficiently using sparse matrix techniques and is therefore applicable to large-scale systems too. By a numerical example, it is demonstrated that the suggested two-step approach has very good approximation capabilities in the time as well as in the frequency domain and can help to reduce the computation time of a numerical simulation significantly.     

A method for improving dynamic solutions in flexible multibody dynamics

This paper presents a method for improving dynamic solutions that are obtained from the dynamic simulation of flexible multibody systems. The mode-acceleration concept in linear structural dynamics is utilized in the proposed method for improving accuracy in the postprocessing stage. A theoretical explanation is made on why the proposed method improves the dynamic solutions in the context of the mode-acceleration method. A mode-acceleration equation for each flexible body is defined and the load term in the right hand side of the equation is represented as a combination of space-dependent and time-dependent terms so that efficient computation of dynamic solutions can be achieved. The load term is obtained from dynamic simulation of a flexible multibody system and a finite element method is used to compute dynamic solutions by quasi-static analyses. Numerical examples show the effectiveness of the proposed method.