Guest Editorial

J. Comput. Nonlinear Dynam. 2013;9(1):010301-010301-1. doi:10.1115/1.4026014.

Multibody system dynamics enables the rapid and economical exploration of machine dynamic behaviors. Multibody dynamics can reduce or even eliminate the need for physical prototyping; accelerate the product development cycle; and reduce development costs. Furthermore, the approach makes it possible to virtually and safely investigate dangerous or destructive events. Often in multibody system dynamics, the analyst assumes mathematically rigid interconnected bodies, which is an acceptable simplification for the analysis of motion and forces in some engineering problems. In an increasing number of cases, however, the deformation of one or more bodies must be considered to achieve the desired simulation accuracies.

Commentary by Dr. Valentin Fuster

Research Papers

J. Comput. Nonlinear Dynam. 2013;9(1):011001-011001-10. doi:10.1115/1.4025277.

In this investigation, a numerical procedure for modeling sliding and nonsliding joint constraints for the B-spline thin plate element is developed for the large deformation analysis of multibody systems. A concept of intermediate reference coordinates proposed for the absolute nodal coordinate formulation is generalized for B-spline elements such that a wide variety of joint constraints can be modeled using existing joint constraint libraries already implemented in multibody dynamics codes. This procedure allows for modeling sliding joints for B-spline elements that requires a solution to moving boundary problems by introducing time-variant surface parameters in the B-spline parametric domain. Since surface parameters treated as knot variables in the basis function are defined in the entire parametric domain rather than the element domain, the location of the constraint definition point can be determined without knowing in which elements the sliding point is located. Furthermore, using the B-spline recurrence formula, control points used for describing the constraint equations can be systematically extracted. It is shown that many types of nonsliding joints fixed on the flexible body can also be modeled as a special case of the sliding joint formulation developed in this investigation, leading to a unified joint constraint formulation for B-spline elements. Several numerical examples are presented in order to demonstrate the use of the numerical procedure developed in this investigation.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;9(1):011002-011002-10. doi:10.1115/1.4025280.

The development of a multibody model of a motorbike L-twin engine cranktrain is presented in this work. The need for an accurate evaluation of the loads acting on the main engine components at high rotational speed makes it necessary to take element flexibility into account in order to capture elastodynamic effects, which might have a major impact on the dynamics of the system. Starting from finite element descriptions of both the crankshaft and the connecting rod, the classical Craig–Bampton (CB) technique is employed to obtain reduced models, which are suitable for the subsequent multibody analysis. A particular component mode selection procedure is implemented based on the concept of effective interface mass, allowing an assessment of the accuracy of the reduced model prior to the nonlinear simulation phase. Bearing dynamics also plays an important role in such a high-speed engine application: angular contact ball bearings are modeled according to a 5DOF nonlinear scheme in order to grasp the main bearings behavior while an impedance-based hydrodynamic bearing model is implemented providing an enhanced operation prediction at big end locations. The assembled cranktrain model is simulated using a commercial multibody software platform. Numerical results demonstrate the effectiveness of the procedure implemented for the flexible component model reduction. The advantages of this technique over the traditional mode truncation approach are discussed.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;9(1):011003-011003-10. doi:10.1115/1.4025285.

Service robots must comply with very demanding safety requirements in order to guarantee that a human can be assisted without any risk of injury. This paper presents a detailed multibody model of the interaction between a single link manipulator and a human head–neck to study the different and more significant parameters involved in the design of the manipulator. The multibody model is first validated through comparison with experimental results obtained in a testbed, which has been built for this purpose. The testbed consists of a flexible pendulum with an inertial wheel attached to the pendulum shaft and a head–neck dummy of 1 degree of freedom (DOF). A phenomenological model of the robot-arm foam soft cover has been developed by fitting experimental results obtained in a compressive test performed on the foam. Once the multibody model is qualitatively validated, several simulations are carried out. The aim of the simulations is to study the effect of different design parameters in the head injury. In particular, the effects of the link flexibility, of the joint compliance, and of the soft cover are detailed.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;9(1):011004-011004-12. doi:10.1115/1.4025352.

This paper presents an approach to control design for flexible structures based on the transfer matrix method (TMM). The approach optimizes the closed-loop pole locations while working directly on the infinite-dimensional TMM model. The approach avoids spatial discretization, eliminating the possibility of modal spillover. The design strategy is based on an iterative process of optimizing the closed-loop pole locations using a Nelder-Mead simplex algorithm and then performing hardware-in-the-loop experiments to see how the pole locations are affecting the closed-loop step response. The evolution of the cost function used to optimized the pole locations is discussed. Contour plots (three dimensional Bode plots) in the complex s-plane are used to visualize the pole locations. A computationally efficient methodology for finding the closed-loop pole locations during the optimization is presented. The technique is applied to a single-flexible-link robot and experimental results show that the optimization procedure improves upon an initial, Bode-based compensator design, leading to a lower settling time.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;9(1):011005-011005-10. doi:10.1115/1.4025284.

In order to model a long flexible body subjected to a moving load within multibody systems, the flexibility can be considered by using a special floating frame of reference approach. In this approach the body deformations are described using shape functions defined in a frame of reference that follows the load. The definition of the deformation shape functions in the load-following frame of reference leads to additional terms of the inertia forces of the flexible body. This method was recently presented by the authors and named the moving modes method. The selected shape functions used in this work are the steady deformation shown by a flexible straight body subjected to a moving load. In this investigation the new formulation is applied to the steady motion and stability analysis of railroad vehicles moving on curved tracks.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;9(1):011006-011006-9. doi:10.1115/1.4025281.

This paper sets out to demonstrate three things: (i) implicit integration with absolute nodal coordinate formulation (ANCF) is effective in handling very stiff systems when an accurate computation of the sensitivity matrix is part of the solution sequence, (ii) parallel computing can provide a vehicle for ANCF to tackle very large kinematically constrained problems with millions of degrees of freedom and produce results in a matter of seconds, and (iii) large systems of equations associated with implicit integration can be solved in parallel by relying on an iterative approach that avoids costly matrix factorizations, which would be prohibitively expensive and memory intensive. For (iii), the approach adopted relies on a Krylov–subspace method that is invoked in the Newton stage at each time step of the numerical solution process. The proposed approach is validated against a commercial package and several simple systems for which analytical solutions are available. A set of numerical experiments demonstrates the scaling of the parallel solution method and provides insights in relation to the size of ANCF problems that are tractable using graphics processing unit (GPU) parallel computing and implicit numerical integration.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;9(1):011007-011007-10. doi:10.1115/1.4025283.

This work presents a flexible multibody system model for the inside turning of thin-walled cylinders. The model accounts for the varying input and output behavior of the workpiece due to workpiece rotation and tool feed, as well as the changing workpiece dynamics due to material removal. A parametric approach is used to incorporate the effect of material removal. Hereby, a number of systems are precalculated for different machined states that are then interpolated to obtain the model for the desired machined state. As different systems typically have different vectors of degrees of freedom, a preprocessing step must be added to guarantee compatibility and thus a meaningful interpolation. In this work, two ways of obtaining a parametric model are presented that lead to similar results. The parametric model is then used to analyze stability of an inside turning operation. By taking into account the varying workpiece dynamics, an improved tool feed is suggested that would allow to greatly reduce the cycle time.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;9(1):011008-011008-7. doi:10.1115/1.4025353.

The modeling and simulation of flexible multibody systems containing fluid-conveying pipes are considered. It is assumed that the mass-flow rate is prescribed and constant and the pipe cross section is piecewise uniform. An existing beam element capable of handling large motions is modified to include the effect of the fluid flow and the initial curvature of the pipe. The modified element is incorporated in a finite-element based multibody system dynamics program, which takes care of the connection with other parts of the system and the simulation. The element is applied in several test problems: the buckling of a simply supported pipe, the flutter instability of a cantilever pipe, and the motion of a curved pipe that can rotate about an axis perpendicular to its plane. As a three-dimensional example, a Coriolis mass-flow rate meter with a U-shaped pipe is considered.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;9(1):011009-011009-8. doi:10.1115/1.4025505.

Experimental monitoring of dynamic response is generally limited to few locations in the system. However, the analysis of structural performance and design of control systems would benefit from a complete knowledge of the system dynamic during service. A numerical approach is developed to numerically reconstruct the load and response of a complete structure from few reference points, based on a modal approach for projecting the response at few points on the domain of the structure. This methodology is particularly advantageous when full-field monitoring of a structure is not a possible solution. An assembly of two beams joined by a nonlinear torsional spring is analyzed in case of different load distributions acting on its span. The approach is shown to be robust and reliable.

Topics: Sensors , Stress , Algorithms
Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;9(1):011010-011010-11. doi:10.1115/1.4025278.

Thin web materials are commonly transported through machinery where a process adds value to the web. The flexible web is supported intermittently by contact with rollers. The friction forces associated with this contact are largely responsible for determining the lateral mechanics and dynamics of the thin web transiting rollers in roll-to-roll process machinery. The investigation focuses on cases where slippage between the rollers and web has become substantial and has resulted in a complex lateral behavior of the web. Two methods are presented for investigating the frictional forces and the resulting lateral behavior. The first method employs explicit finite element (FE) dynamic analysis to study the lateral mechanics of the web after steady state behavior has been achieved. This method allows the direct study of the frictional forces. The second method employs Laser Doppler Velocimetry in a novel experimental noncontact technique to examine internal loads within the web, which were influenced by the frictional forces. Both methods are shown to provide results, which agree with one another and with previous analysis. The analyses are used to form a new friction boundary condition between a web and roller that will benefit other analysis methods.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;9(1):011011-011011-9. doi:10.1115/1.4025351.

Rack feeders for the automated operation of high bay rackings are of high practical importance. They are characterized by a horizontally movable carriage supporting a tall and flexible vertical beam structure, on which a cage containing the payload can be positioned in vertical direction. To shorten the transport times by using trajectories with increased maximum acceleration and jerk values, accompanying control measures can be introduced counteracting or avoiding undesired vibrations of the flexible structure. In this contribution, both the control-oriented modeling for an experimental setup of such a flexible rack feeder and the model-based design of a gain-scheduled feedforward and feedback control structure are presented. Whereas, a kinematical model is sufficient for the vertical axis, the horizontal motion of the rack feeder is modeled as a planar elastic multibody system with the cage position as scheduling parameter. For the mathematical description of the bending deflections, a one-dimensional Ritz ansatz is introduced. The tracking control design is performed separately for both the horizontal and the vertical axes using decentralized state-space representations. Remaining model uncertainties are estimated by a disturbance observer. The resulting tracking accuracy of the proposed control concept is shown by measurement results from the experimental setup. Furthermore, these results are compared to those obtained with an alternative control concept from previous work.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;9(1):011012-011012-10. doi:10.1115/1.4025279.

Current challenges in industrial multibody system simulation are often beyond the classical range of application of existing industrial simulation tools. The present paper describes an extension of a recursive order-n multibody system (MBS) formulation to nonlinear models of flexible deformation that are of particular interest in the dynamical simulation of wind turbines. The floating frame of reference representation of flexible bodies is generalized to nonlinear structural models by a straightforward transformation of the equations of motion (EoM). The approach is discussed in detail for the integration of a recently developed discrete Cosserat rod model representing beamlike flexible structures into a general purpose MBS software package. For an efficient static and dynamic simulation, the solvers of the MBS software are adapted to the resulting class of MBS models that are characterized by a large number of degrees of freedom, stiffness, and high frequency components. As a practical example, the run-up of a simplified three-bladed wind turbine is studied where the dynamic deformations of the three blades are calculated by the Cosserat rod model.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;9(1):011013-011013-8. doi:10.1115/1.4025282.

For the modeling of large deformations in multibody dynamics problems, the absolute nodal coordinate formulation (ANCF) is advantageous since in general, the ANCF leads to a constant mass matrix. The proposed ANCF beam finite elements in this approach use the transverse slope vectors for the parameterization of the orientation of the cross section and do not employ an axial nodal slope vector. The geometric description, the degrees of freedom, and a continuum-mechanics-based and a structural-mechanics-based formulation for the elastic forces of the beam finite elements, as well as their usage in several static problems, have been presented in a previous work. A comparison to results provided in the literature to analytical solution and to the solution found by commercial finite element software shows accuracy and high order convergence in statics. The main subject of the present paper is to show the usability of the beam finite elements in dynamic and buckling applications.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;9(1):011014-011014-9. doi:10.1115/1.4025476.

The inverse dynamics of flexible multibody systems is formulated as a two-point boundary value problem for an index-3 differential-algebraic equation (DAE). This DAE represents the equation of motion with kinematic and trajectory constraints. For so-called nonminimum phase systems, the remaining dynamics of the inverse model is unstable. Therefore, boundary conditions are imposed not only at the initial time but also at the final time in order to obtain a bounded solution of the inverse model. The numerical solution strategy is based on a reformulation of the DAE in index-2 form and a multiple shooting algorithm, which is known for its robustness and its ability to solve unstable problems. The paper also describes the time integration and sensitivity analysis methods that are used in each shooting phase. The proposed approach does not require a reformulation of the problem in input-output normal form, which is known from nonlinear control theory. It can deal with serial and parallel kinematic topology, minimum phase and nonminimum phase systems, and rigid and flexible mechanisms.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;9(1):011015-011015-10. doi:10.1115/1.4025627.

Static equations for thin inextensible elastic rods, or elastica as they are sometimes called, have been studied since before the time of Euler. In this paper, we examine how to model the dynamic behavior of elastica. We present a fairly high speed, robust numerical scheme that uses (i) a space discretization that uses cubic splines, and (ii) a time discretization that preserves a discrete version of the Hamiltonian. A good choice of numerical scheme is important because these equations are very stiff; that is, most explicit numerical schemes will become unstable very quickly. The authors conducted this research anticipating describing the dynamic Kirchhoff problem, that is, the behavior of general springs that have natural curvature, and for which the equations take into account torsion of the rod.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;9(1):011016-011016-10. doi:10.1115/1.4025669.

Flexure hinges inherently lose stiffness in supporting directions when deflected. In this paper a method is presented for optimizing the geometry of flexure hinges, which aims at maximizing supporting stiffnesses. In addition, the new -flexure hinge design is presented. The considered hinges are subjected to a load and deflected an angle of up to ±20 deg. The measure of performance is defined by the first unwanted natural frequency, which is closely related to the supporting stiffnesses. During the optimization, constraints are applied to the actuation moment and the maximum occurring stress. Evaluations of a curved hinge flexure, cross revolute hinge, butterfly flexure hinge, two cross flexure hinge types, and the new -flexure hinge are presented. Each of these hinge types is described by a parameterized geometric model. A flexible multibody modeling approach is used for efficient modeling while it accounts for the nonlinear geometric behavior of the stiffnesses. The numerical efficiency of this model is very beneficial for the design optimization. The obtained optimal hinge designs are validated with a finite element model and show good agreement. The optimizations show that a significant increase in supporting stiffness, with respect to the conventional cross flexure hinge, can be achieved with the -flexure hinge.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;9(1):011017-011017-9. doi:10.1115/1.4025635.

In many cases, the design of a tracking controller can be significantly simplified by the use of a 2-degrees of freedom (DOF) control structure, including a feedforward control (i.e., the inversion of the nominal system dynamics). Unfortunately, the computation of this feedforward control is not easy if the system is nonminimum-phase. Important examples of such systems are flexible multibody systems, such as lightweight manipulators. There are several approaches to the numerical computation of the exact inversion of a flexible multibody system. In this paper, the singularly perturbed form of such mechanical systems is used to give a semianalytic solution to the tracking control design. The control makes the end-effector to even though not exactly, but approximately track a certain trajectory. Thereby, the control signal is computed as a series expansion in terms of an overall flexibility of the bodies of the multibody system. Due to the use of symbolic computations, the main calculations are independent of given parameters (e.g., the desired trajectories), such that the feedforward control can be calculated online. The effectiveness of this approach is shown by the simulation of a two-link flexible manipulator.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;9(1):011018-011018-11. doi:10.1115/1.4026059.

The implementation of flexible instruments in surgery necessitates high motion and force fidelity and good controllability of the tip. However, the positional accuracy and the force transmission of these instruments are jeopardized by the friction, the clearance, and the inherent compliance of the instrument. The surgical instrument is modeled as a series of interconnected spatial beam elements. The endoscope is modeled as a rigid curved tube. The stiffness, damping, and friction are defined in order to calculate the interaction between the instrument and the tube. The effects of various parameters on the motion and force transmission behavior were studied for the axially-loaded and no-load cases. The simulation results showed a deviation of 1.8% in the estimation of input force compared with the analytical capstan equation. The experimental results showed a deviation on the order of 1.0%. The developed flexible multibody model is able to demonstrate the characteristic behavior of the flexible instrument for both the translational and rotational input motion for a given set of parameters. The developed model will help us to study the effects of various parameters on the motion and force transmission of the instrument.

Commentary by Dr. Valentin Fuster

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