A frequency domain approach for parameter identification in multibody dynamics

The adjoint method shows an efficient way to incorporate inverse dynamics to engineering multibody applications, as, e.g., parameter identification. In case of the identification of parameters in oscillating multibody systems, a combination of Fourier analysis and the adjoint method is an obvious and promising approach. The present paper shows the adjoint method including adjoint Fourier coefficients for the parameter identification of the amplitude response of oscillations. Two examples show the potential and efficiency of the proposed method in multibody dynamics.     

Comparing double-step and penalty-based semirecursive formulations for hydraulically actuated multibody systems in a monolithic approach

The simulation of mechanical systems often requires modeling of systems of other physical nature, such as hydraulics. In such systems, the numerical stiffness introduced by the hydraulics can become a significant aspect to consider in the modeling, as it can negatively effect to the computational efficiency. The hydraulic system can be described by using the lumped fluid theory. In this approach, a pressure can be integrated from a differential equation in which effective bulk modulus is divided by a volume size. This representation can lead to numerical stiffness as a consequence of which time integration of a hydraulically driven system becomes cumbersome. In this regard, the used multibody formulation plays an important role, as there are many different procedures for the constraint enforcement and different sets of coordinates to choose from. This paper introduces the double-step semirecursive approach and compares it with a penalty-based semirecursive approach in case of coupled multibody and hydraulic dynamics within the monolithic framework. To this end, hydraulically actuated four-bar and quick-return mechanisms are analyzed as case studies. The two approaches are compared in terms of the work cycle, energy balance, constraint violation, and numerical efficiency of the mechanisms. It is concluded that the penalty-based semirecursive approach has a number of advantages compared with the double-step semirecursive approach, which is in accordance with the literature.

     

Dynamic Simulation of Multibody Systems Using a New State-Time Methodology

This paper presents a new methodology demonstrating the feasibility and advantages of a state-time formulation for dynamic simulation of complex multibody systems which shows potential advantages for exploiting massively parallel computing resources. This formulation allows time to be discretized and parameterized so that it can be treated as a variable in a manner similar to the system state variables. As a consequence of such a state-time discretization scheme, the system of governing equations yields to a set of loosely coupled linear-quadratic algebraic equations that is well-suited in structure for some families of nonlinear algebraic equations solvers. The goal of this work is to develop efficient multibody dynamics algorithm that is extremely scalable and better able to fully exploit anticipated immensely parallel computing machines (tera flop, peta flop and beyond) made available to it.     

A rotationless formulation of multibody dynamics: Modeling of screw joints and incorporation of control constraints

In the present work, a new energy-momentum conserving time-stepping scheme for multibody systems comprising screw joints is developed. In particular, it is shown that the underlying rotationless formulation of multibody dynamics along with a specific coordinate augmentation technique makes possible the energy-momentum discretization of the screw pair. In addition to that, control (or servo) constraints are treated within the rotationless framework of multibody dynamics. The control constraints are used to partially prescribe the motion of a multibody system. In particular, control constraints, in conjunction with the coordinate augmentation technique, make possible to solve inverse dynamics problems by applying the present simulation approach.     

Structural analysis of the intervertebral discs adjacent to an interbody fusion using multibody dynamics and finite element cosimulation

This work describes a methodology for the dynamic and structural analysis of complex (bio)mechanical systems that joins both multibody dynamics and finite element domains, in a synergetic way, through a cosimulation procedure that takes benefit of the advantages of each numerical formulation. To accomplish this goal, a cosimulation module is developed based on the gluing algorithm X-X, which is the key element responsible for the management of the information flux between the two software packages (each using its own mathematical formulation and code). The X-X algorithm uses for each cosimulated structure multiple pairs of reference points whose kinematics are solved by the multibody module and prescribed, as initial data, to the finite element counterpart. The finite element module, by its turn, solves the structural problem imposed by the prescribed kinematics, calculates the resulting generalized loads applied over the reference points and return these loads back to the multibody module that uses them to solve the dynamic problem and to calculate new reference kinematics to prescribe to the finite element module in the next time step. The proposed method is applied to study the cervical spine dynamics in a pathologic situation in which an intersomatic fusion is simulated to confirm its potential advantages. Taking into account the proposed simulation scenario, a cervical spine multibody model that includes the rigid vertebrae, the facet joints’ and spinous processes’ contacts, ligaments and the finite element models of the intervertebral discs, and their surrogates is developed. The proposed model is simulated for extension in a forward dynamics perspective.     

Modeling and simulation of structural components in recursive closed-loop multibody systems

Passenger cars, transit buses, railroad vehicles, off-highway trucks, earth moving equipment and construction machinery contain structural and light-fabrications (SALF) components that are prone to excessive vibration due to rough terrains and work-cycle loads’ excitations. SALF components are typically modeled as flexible components in the multibody system allowing the analysts to predict elastic deformation and hence the stress levels under different loading conditions. Including SALF component in the multibody system typically generates closed-kinematic loops. This paper presents an approach for integrating SALF modeling capabilities as a flexible body in a general-purpose multibody dynamics solver that is based on joint-coordinates formulation with the ability to handle closed-kinematic loops. The spatial algebra notation is employed in deriving the spatial multibody dynamics equations of motion. The system kinematic topology matrix is used to project the Cartesian quantities into the joint subspace, leading to a condensed set of nonlinear equations with minimum number of generalized coordinates. The proposed flexible body formulation utilizes the component mode synthesis approach to reduce the large number of finite element degrees of freedom to a small set of generalized modal coordinates. The resulting reduced flexible body model has two main characteristics: the stiffness matrix is constant while the mass matrix depends on the elastic modal coordinates. A consistent set of pre-computed inertia shape integrals are identified and used to update the modal mass matrix at each time step. The implementation of the component mode synthesis approach in a closed-loop recursive multibody formulation is presented. The kinematic equations are modified to include the effect of the flexible body modal elastic coordinates. Also, modified constraint equations that include the effect of flexibility at the joint connections and the necessary details of the Jacobian matrix are presented. Baumgarte stabilization approach is used to stabilize the constraint equations without using iterative schemes. A sample results for flexible body impeded in a closed system will be presented to demonstrate the above mentioned approach.     

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.     

Modular and Recursive Kinematics and Dynamics for Parallel Manipulators

Constrained multibody systems typically feature multiple closed kinematic loops that constrain the relative motions and forces within the system. Typically, such systems possess far more articulated degrees-of-freedom (within the chains) than overall end-effector degrees-of-freedom.Thus, actuation of a subset of the articulations creates mixture of active and passive joints within the chain.The presence of such passive joints interferes with the effective modular formulation of the dynamic equations-of-motion in terms of a minimal set of actuator coordinates as well the subsequent recursivesolution for both forward and inverse dynamics applications. Thus, in this paper, we examine the development of modular and recursive formulations of equations-of-motion in terms of a minimal set of actuated-joint-coordinates for an exactly-actuated parallel manipulators. The 3 RRR planar parallel manipulator, selected to serve as a case-study, is an illustrative example of a multi-loop, multi-degree-of-freedom system with mixtures of active/passive joints. The concept of decoupled natural orthogonal complement (DeNOC) is combined with the spatial parallelism inherent in parallel mechanisms to develop a dynamics formulation that is both recursive and modular. An algorithmic approach to the development of both forward and inverse dynamics is highlighted. The presented simulation studies highlight the overall good numerical behavior of the developed formulation, both in terms of accuracy and lack of formulation stiffness.     

Modular Simulation in Multibody System Dynamics

The dynamic analysis of complex engineering systems likeautomobiles is often relieved by a modular approach to make it treatableby a team of engineers. The modular decomposition is based onengineering intuition of corresponding engineering disciplines. In thispaper, a modular formulation of multibody systems is proposed which isbased on the block representation of a multibody system withcorresponding input and output quantities. Advantages of this modularapproach range from independent and parallel modeling of subsystems overthe easy exchange of the resulting modules to the use of differentsoftware for each module. However, the modular simulation of the globalsystem by coupling of simulators may result in an unstable integrationif an algebraic loop exists between the subsystems. This numericalphenomenon is analyzed and a method of simulator coupling whichguarantees stability for general systems including algebraic loops isintroduced. Numerical results of the modular simulation of aslider-crank mechanism are presented.     

Interaction of flexible multibody systems with fluids analyzed by means of smoothed particle hydrodynamics

The present work deals with a computational approach to fluid-structure interaction (FSI) problems by coupling of flexible multibody system dynamics and fluid dynamics. Since the methods for the numerical modeling are well known, both for the structural and the fluid part, the focus of this work lies on the coupling formalism. Moreover, the applicability of the presented approach to arbitrary geometries and high structural stiffness is studied, as well as an easy model setup. No restriction should be made on the topology of the structure or the complexity of motion.For the fluid part a meshless method, known as smoothed particle hydrodynamics (SPH) is applied, which fulfills the above requirements. While an explicit time integration scheme in SPH provides a fast simulation of the fluid dynamics, advanced methods from flexible multibody dynamics provide a variety of benefits for the simulation of the solid part. Amongst these are specialized structural finite elements for both small and large deformation bodies, joints, stable implicit time-integration schemes, and model reduction techniques.A rule for the interaction between fluids and structures is derived from imposing a distributed potential over boundary segments of the structures, which the fluid particles respond to. The work is concluded by illustrative examples, demonstrating the successful coupling of flexible multibody systems with fluids.     

Discrete time transfer matrix method for dynamics analysis of complex weapon systems

Efficient, precise dynamic modeling and analysis for complex weapon systems have become more and more important in their dynamic design and performance optimizing. As a new method developed in recent years, the discrete time transfer matrix method of multibody system is highly efficient for multibody system dynamics. In this paper, taking a shipboard gun system as an example, by deducing some new transfer equations of elements, the discrete time transfer matrix method of multibody sys- tem is used to solve ...     

Weakly and fully coupled methods for structural optimization of flexible mechanisms

The paper concerns a detailed comparison between two optimization methods that are used to perform the structural optimization of flexible components within a multibody system (MBS) simulation. The dynamic analysis of flexible MBS is based on a nonlinear finite element formulation. The first method is a weakly coupled method, which reformulates the dynamic response optimization problem in a two-level approach. First, a rigid or flexible MBS simulation is performed, and second, each component is optimized independently using a quasi-static approach in which a series of equivalent static load (ESL) cases obtained from the MBS simulation are applied to the respective components. The second method, the fully coupled method, performs the dynamic response optimization using the time response obtained directly from the flexible MBS simulation. Here, an original procedure is proposed to evaluate the ESL from a nonlinear finite element simulation, contrasting with the floating reference frame formulation exploited in the standard ESL method. Several numerical examples are provided to support our position. It is shown that the fully coupled method is more general and accommodates all types of constraints at the price of a more complex optimization process.     

Three Approaches for Elastodynamic Contact in Multibody Systems

In machine dynamics impacts may occur by interaction of solid bodies. There is no doubt that the method of multibody systems is most efficient for the dynamical analysis of the overall motion. However, during impact energy is lost macromechanically measured by the coefficient of restitution. This coefficient has to be estimated from experiments and experience but cannot be computed within the multibody system approach. The impacts, on the other hand, are generating waves in the bodies which are propagating until they are vanishing due to material damping. These high frequency phenomena are analyzed using wave propagation, modal approach and finite elements. The results of the simulation on the fast time scale are used to compute the coefficient of restitution which is then fed back to the multibody system equations and the solution continues on the related slow time scale. The efficiency of the approach presented is shown for the impact of a steel sphere on four different shaped aluminum bodies of comparable mass. The simulation results are verified with experiments performed on different time scales, too. Using the impact of a double pendulum on a stop as example, the application of the multiscale approach to a multibody system is shown.     

Penalty,Semi-Recursive and Hybrid Methods for MBS Real-Time Dynamics in the Context of Structural Integrators

In a previous work, the authors presented a new real-time formulation for the dynamics of multibody systems, which encompasses high ranks of efficiency, accuracy, robustness and easiness of implementation. The new method, called hybrid, was obtained as a combination of a topological semi-recursive formulation based on velocities transformation, and a global penalty formulation for closed-loops consideration. It was proven to be more robust and efficient that its predecessors for large problems. For the three methods compared, the implicit, single-step trapezoidal rule was used as numerical integrator. In this paper, the influence of the integration scheme on the performance of the three mentioned methods is studied. Since the hybrid formulation becomes competitive for large multibody systems, a rather demanding simulation of the full model of a car vehicle is selected as benchmark problem. Computer codes implementing the three dynamic formulations in combination with different structural integrators, like Newmark, HHT and Generalized- algorithms, are used to run the simulation, so that the performance of each couple dynamic-formulation/numerical-integrator can be appraised. The example is also analyzed through a commercial tool, so as to provide the readers with a well-known reference for comparison.     

Modeling Multibody Systems with Uncertainties. Part I: Theoretical and Computational Aspects

This study explores the use of generalized polynomial chaos theory for modeling complex nonlinear multibody dynamic systems in the presence of parametric and external uncertainty. The polynomial chaos framework has been chosen because it offers an efficient computational approach for the large, nonlinear multibody models of engineering systems of interest, where the number of uncertain parameters is relatively small, while the magnitude of uncertainties can be very large (e.g., vehicle-soil interaction). The proposed methodology allows the quantification of uncertainty distributions in both time and frequency domains, and enables the simulations of multibody systems to produce results with “error bars”. The first part of this study presents the theoretical and computational aspects of the polynomial chaos methodology. Both unconstrained and constrained formulations of multibody dynamics are considered. Direct stochastic collocation is proposed as less expensive alternative to the traditional Galerkin approach. It is established that stochastic collocation is equivalent to a stochastic response surface approach. We show that multi-dimensional basis functions are constructed as tensor products of one-dimensional basis functions and discuss the treatment of polynomial and trigonometric nonlinearities. Parametric uncertainties are modeled by finite-support probability densities. Stochastic forcings are discretized using truncated Karhunen-Loeve expansions. The companion paper “Modeling Multibody Dynamic Systems With Uncertainties. Part II: Numerical Applications” illustrates the use of the proposed methodology on a selected set of test problems. The overall conclusion is that despite its limitations, polynomial chaos is a powerful approach for the simulation of multibody systems with uncertainties.     

A 3D Finite Element Method for Flexible Multibody Systems

An efficient finite element (FE) formulation for the simulation of multibody systems is derived from Hamilton's principle. According to the classical assumptions of multibody systems, a large rotation formulation has been chosen, where large rotations and large displacements, but only small deformations of the single bodies are taken into account. The strain tensor is linearized with respect to a co-rotated frame. The present approach uses absolute coordinates for the degrees of freedom and forms an alternative to the floating frame of reference formulation that is based on relative coordinates and describes deformation with respect to a co-rotated frame. Due to the modified strain tensor, the present formulation distinguishes significantly from standard nodal based nonlinear FE methods. Constraints are defined in integral form for every pair of surfaces of two bodies. This leads to a small number of constraint equations and avoids artificial stress singularities. The resulting mass and stiffness matrices are constant apart from a transformation based on a single rotation matrix for each body. The particular structure of this transformation allows to prevent from the usually expensive factorization of the system Jacobian within implicit time--integration methods. The present method has been implemented and tested with the FE-package NGSolve and specific 3D examples are verified with a standard beam formulation.