Flexible Multibody Systems Models Using Composite Materials Components

The use of a multibody methodology to describe the large motion of complex systems that experience structural deformations enables to represent the complete system motion, the relative kinematics between the components involved, the deformation of the structural members and the inertia coupling between the large rigid body motion and the system elastodynamics. In this work, the flexible multibody dynamics formulations of complex models are extended to include elastic components made of composite materials, which may be laminated and anisotropic. The deformation of any structural member must be elastic and linear, when described in a coordinate frame fixed to one or more material points of its domain, regardless of the complexity of its geometry. To achieve the proposed flexible multibody formulation, a finite element model for each flexible body is used. For the beam composite material elements, the sections properties are found using an asymptotic procedure that involves a two-dimensional finite element analysis of their cross-section. The equations of motion of the flexible multibody system are solved using an augmented Lagrangian formulation and the accelerations and velocities are integrated in time using a multi-step multi-order integration algorithm based on the Gear method.     

Performance of automatic differentiation tools in the dynamic simulation of multibody systems

Within the multibody systems literature, few attempts have been made to use automatic differentiation for solving forward multibody dynamics and evaluating its computational efficiency. The most relevant implementations are found in the sensitivity analysis field, but they rarely address automatic differentiation issues in depth. This paper presents a thorough analysis of automatic differentiation tools in the time integration of multibody systems. To that end, a penalty formulation is implemented. First, open-chain generalized positions and velocities are computed recursively, while using Cartesian coordinates to define local geometry. Second, the equations of motion are implicitly integrated by using the trapezoidal rule and a Newton–Raphson iteration. Third, velocity and acceleration projections are carried out to enforce kinematic constraints. For the computation of Newton–Raphson’s tangent matrix, instead of using numerical or analytical differentiation, automatic differentiation is implemented here. Specifically, the source-to-source transformation tool ADIC2 and the operator overloading tool ADOL-C are employed, in both dense and sparse modes. The theoretical approach is backed with the numerical analysis of a 1-DOF spatial four-bar mechanism, three different configurations of a 15-DOF multiple four-bar linkage, and a 16-DOF coach maneuver. Numerical and automatic differentiation are compared in terms of their computational efficiency and accuracy. Overall, we provide a global perspective of the efficiency of automatic differentiation in the field of multibody system dynamics.     

A linearized input–output representation of flexible multibody systems for control synthesis

In this paper, a linearized input–output representation of flexible multibody systems is proposed in which an arbitrary combination of positions, velocities, accelerations, and forces can be taken as input variables and as output variables. The formulation is based on a nonlinear finite element approach in which a multibody system is modeled as an assembly of rigid body elements interconnected by joint elements such as flexible hinges and beams. The proposed formulation is general in nature and can be applied for prototype modeling and control system analysis of mechatronic systems. Application of the theory is illustrated through a detailed model development of an active vibration isolation system for a metrology frame of a lithography machine.     

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.     

An Efficient Method for Synthesis of Planar Multibody Systems Including Shape of Bodies as Design Variables

A point contact joint has been developed and implemented in a joint coordinate based planar multibody dynamics analysis program that also supports revolute and translational joints. Further, a segment library for the definition of the contours of the point contact joints has been integrated in the code and as a result any desired contour shape may be defined. The sensitivities of the basic physical variables of a multibody system, i.e., the positions, velocities, accelerations and reactions of the system with respect to the automatically identified independent design variables may be determined analytically, allowing design problems where the shape of the bodies are of interest to be handled in both a general and efficient way.     

Structural topology optimization of multibody systems

Flexible multibody dynamics (FMD) has found many applications in control, analysis and design of mechanical systems. FMD together with the theory of structural optimization can be used for designing multibody systems with bodies which are lighter, but stronger. Topology optimization of static structures is an active research topic in structural mechanics. However, the extension to the dynamic case is less investigated as one has to face serious numerical difficulties. One way of extending static structural topology optimization to topology optimization of dynamic flexible multibody system with large rotational and transitional motion is investigated in this paper. The optimization can be performed simultaneously on all flexible bodies. The simulation part of optimization is based on an FEM approach together with modal reduction. The resulting nonlinear differential-algebraic systems are solved with the error controlled integrator IDA (Sundials) wrapped into Python environment by Assimulo (Andersson et al. in Math. Comput. Simul. 116(0):26–43, 2015). A modified formulation of solid isotropic material with penalization (SIMP) method is suggested to avoid numerical instabilities and convergence failures of the optimizer. Sensitivity analysis is central in structural optimization. The sensitivities are approximated to circumvent the expensive calculations. The provided examples show that the method is indeed suitable for optimizing a wide range of multibody systems. Standard SIMP method in structural topology optimization suggests stiffness penalization. To overcome the problem of instabilities and mesh distortion in the dynamic case we consider here additionally element mass penalization.     

A time-scaling method for near-time-optimal control of an omni-directional robot along specified paths

This paper proposes a time-scaling method to determine the near-time-optimal movement of an omnidirectional mobile robot along a given reference path. With this strategy, the positions of the trajectory after scaling are the same as the original ones such that the geometric path constraints are not violated. However, the velocities and the accelerations are adjusted to meet the dynamical constraints and to minimize the traveling time. When determining the time-scaling function, a cubic spline interpolation technique is used, in which control points for interpolation are determined simultaneously by a particle swarm optimization (PSO) method based on the integration of a time-scaling function. To show the feasibility of the proposed method, the results of a simulation example is illustrated. This work was presented in part at the 13th International Symposium on Artificial Life and Robotics, Oita, Japan, January 31–February 2, 2008     

A practical method for the deformation of long-stroke hydraulic manipulators in grasping-handling tasks

Long-stroke hydraulic manipulators are utilized in various grasping-handling tasks, and the flexible deformation of these manipulators is the primary obstacle that affects precise position control of the end-effectors in Cartesian space. This deformation is manifested in the following three aspects: joint deformation, structural deformation and clearance variation. Due to deformation uncertainty, methods that model the hydraulic manipulator as a combination of flexible multibody systems and hydraulic actuators are unsuitable. In this article, we propose an incremental inverse kinematics model (IIKM) as a new approach to solving the above deformation difficulties. The projection method is used to obtain the inverse kinematic analytical solution of long-stroke hydraulic manipulators, which is based on the manipulator deformation in the current configuration (current configuration refers to the arrangement of the manipulator links when the manipulator starts to move to the target position). The proposed method avoids complex flexible multibody modeling and parameter identification, allowing long-stroke hydraulic manipulators to be accurately controlled within a certain neighborhood. An evaluation coefficient is proposed to analyze the calculation accuracy of the IIKM in combination with the success rate obtained from 190 grasping experiments. Through these experiments, we determine the optimal calculation height range of the IIKM in the vertical direction and the optimal calculation position area in the horizontal direction and prove that the IIKM result can guarantee the success of grasping-handling tasks when the end-effector is within the optimal calculation height range.     

Sensitivity of optimal shapes of artificial grafts with respect to flow parameters

The difficulties arising in the numerical solution of PDE-constrained shape optimization problems are manifold. Key ingredients are the optimization strategy and the shape deformation method. Furthermore, the robustness of the optimal shape with respect to simulation parameters is of great interest. In this paper, we consider fluid flows described by the incompressible Navier–Stokes equations. Previous studies on artificial bypass grafts indicated the need for specific constitutive models to account for the non-Newtonian nature of blood; in particular, the constitutive model was shown to affect the solution of the shape optimization problem. We employ a shape optimization framework that couples a finite element solver with quasi-Newton-type optimizers and a Bezier spline shape parametrization. To compute derivatives of the optimal shapes with respect to viscosity, we transform the entire optimization framework by combining the automatic differentiation tools Adifor2 and TAPENADE. We demonstrate the impact of the geometry parametrization and of geometric constraints on the optimization outcome. Finally, we employ the transformed framework to compute the sensitivity of the optimal shape of bypass grafts with respect to kinematic viscosity. The resulting sensitivities predict very accurately the influence of viscosity changes on the optimal shape. The proposed methodology provides a powerful tool to further investigate the necessity of intricate constitutive models by taking derivatives with respect to model parameters.     

Optimization of the Rigid Body Mechanism in a Wobble Plate Compressor

A complex mechanical system is optimized with respect to its performance. The mechanism is a compressor, which is modeled as a multibody system. The optimization is first performed on a simplified 2D model, where it is possible to find analytical sensitivities, and the results indicate that the mechanism can be optimized. Optimization is finally performed with numerical sensitivities, from a full 3D mechanism simulation with 20 bodies, and the results show that the desired change of performance is obtained. For the optimization procedure the SLP method (sequential linear programming) is used with good results, and although the paper deals with optimization of a specific mechanism, the procedure can be modified to treat also other mechanical systems.     

Sensitivity Analysis of Rigid-Flexible Multibody Systems

An important step in the application of automated design techniques to rigid-flexible multibody systems is the calculation of the sensitivities with respect to design variables. Thispaper presents a general formulation for thecalculation of the first order analytical designsensitivities based on the direct differentiationmethod. The analytical sensitivities are comparedwith the numerical results obtained by the finitedifferences method and the accuracy and validity ofboth methods is discussed. Cartesian co-ordinates areused for the dynamic analysis of rigid-flexiblemultibody systems. To reduce the number ofco-ordinates associated with the flexible bodies, thecomponent mode synthesis method is used. Theequations of the sensitivities are obtainedsymbolically and integrated in time simultaneouslywith the dynamic equations. Examples of 2Dsensitivity analysis of the transient response of aslider-crank and of a vehicle with a flexible chassisare presented, and the accuracy and characteristics ofthe sensitivities are analyzed and discussed.     

Structural optimization with fatigue and ultimate limit constraints of jacket structures for large offshore wind turbines

The purpose of the research presented in this paper is to develop and implement an efficient method for analytical gradient-based sizing optimization of a support structure for offshore wind turbines. In the jacket structure optimization of frame member diameter and thickness, both fatigue limit state, ultimate limit state, and frequency constraints are included. The established framework is demonstrated on the OC4 reference jacket with the NREL 5 MW reference wind turbine installed at a deep water site. The jacket is modeled using 3D Timoshenko beam elements. The aero-servo-elastic loads are determined using the multibody software HAWC2, and the wave loads are determined using the Morison equation. Analytical sensitivities are found using both the direct differentiation method and the adjoint method. An effective formulation of the fatigue gradients makes the amount of adjoint problems that needs to be solved independent of the amount of load cycles included in the analysis. Thus, a large amount of time-history loads can be applied in the fatigue analysis, resulting in a good representation of the accumulated fatigue damage. A reduction of 40 % mass is achieved in 23 iterations using the CPLEX optimizer by IBM ILOG, where both fatigue and ultimate limit state constraints are active at the optimum.     

Impulsive synchronization motion in networked open-loop multibody systems

This paper presents a procedure for studying impulsive synchronization motion in networked open-loop multibody systems formulated by Lagrange dynamics. Impulsive motion occurs when the networked systems are physically subject to either direct or indirect impulsive effects, or when subjected to both simultaneously. The impulsive effects are usually caused by impulsive forces or impulsive constraints. The governing equations of networked open-loop multibody systems are developed from Lagrange formulation. The procedure automatically incorporates a preliminary feedback control and the effects of impulsive constraints through its analysis. Some generic criteria on exponential synchronization of the system output with respect to generalized coordinates and its velocities over, respectively, undirected fixed and switching network topologies, are derived analytically. The procedure shows that impulsive synchronization motion in networked open-loop multibody systems can achieve by impulsive constraints strategies. Two examples and simulations are used to demonstrate and validate the analysis procedure.     

Optimization of flexible components of multibody systems via equivalent static loads

An optimization methodology that iteratively links the results of multibody dynamics and structural analysis software to an optimization method is presented to design flexible multibody systems under dynamic loading conditions. In particular, rigid multibody dynamic analysis is utilized to calculate dynamic loads of a multibody system and a structural optimization algorithm using equivalent static loads transformed from the dynamic loads are used to design the flexible components in the multibody dynamic system. The equivalent static loads, which are derived from equations of motion, are used as multiple loading conditions of linear structural optimization. A simple example is solved to verify the proposed methodology and the pelvis part of the biped humanoid, a complex multibody system which consists of many bodies and joints, is redesigned using the proposed methodology.     

Joint Reaction Forces in Multibody Systems with Redundant Constraints

Redundant constraints are defined as those constraints which can be removed without changing the kinematics of the mechanism. They are usually eliminated from the mathematical model of a multibody system. For a given mechanism the set of redundant constraints can be chosen in many ways. Rigid body systems with redundant constraints do not have a unique solution to the problem of joint reaction forces calculation. If redundant constraints are present in the mechanical system, then the system is statically undetermined. If in the case of dynamics problem the constraints are consistent, all of them are frictionless and we are interested only in positions, velocities and accelerations of the bodies, then the calculation of joint reaction forces is not necessary. In many cases, however, e.g. when we want to take into account friction in joints, the calculation of joint reaction forces cannot be avoided. In some rigid body systems, despite the redundant constraints existence, reaction forces in selected joints can be uniquely determined. The paper presents three methods of finding the constraints for which reaction forces can be uniquely determined using rigid body model. Three different techniques of Jacobian matrix analysis are used.     

Equilibrium analysis of multibody dynamic systems using genetic algorithm in comparison with constrained and unconstrained optimization techniques

The present paper describes a set of procedures for the solution of nonlinear equilibrium problems in complex multibody systems. To find the equilibrium position of the system, six different optimization algorithms are used to minimize the total potential energy (TPE) of the system and compared with respect to accuracy and efficiency. A computer program is developed to evaluate the equality constraints and objective function of a general multibody dynamic system to find the equilibrium condition. It is seen that the indirect methods have better results and converge faster. Also it is shown that the genetic algorithm (GA) results in a global optimum while the other methods converge to a local optimum.     

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.     

Dealing with multiple contacts in a human-in-the-loop application

This paper deals with continuous contact force models applied to the human-in-the-loop simulation of multibody systems, while the results are valid in general to all the real-time applications with contacts. The contact model proposed in this work is suited to collisions between massive solids for which the assumption of quasi-static contact holds, and it can be supposed that the deformation is limited to a small region of the colliding bodies while the remainder of them are assumed to be rigid. The model consists of two components: normal compliance with nonlinear viscoelastic model based on the Hertz law, and tangential friction force based on Coulomb’s law including sticktion and a viscous friction component. Furthermore, the model takes into account the geometry and the material of the colliding bodies. The tangential model is a novel contribution while the normal model is completely taken from previous works. For this work, the formulation of the equations of motion is an augmented Lagrangian with projections of velocities and accelerations onto their constraints manifolds and implicit integrator. The whole solution proposed is tested in three applications: the first one is the simulation of a spring–mass system with Coulomb’s friction, which is an academic problem with known analytical solution; the second one is the Bowden and Leben stick–slip experiment; the third one is a simulator of a hydraulic excavator Liebherr A924, which is a realistic application that gives an idea of the capabilities of the method proposed.