FreeDyn

Within the framework of this article, we pursue a novel approach for the determination of time-optimal controls for dynamic systems under observance of end conditions. Such problems arise in robotics, e.g., if the control of a robot has to be designed such that the time for a rest-to-rest maneuver becomes a minimum. So far, such problems have been generally considered as two-point boundary value problems, which are hard to solve and require an initial guess close to the optimal solution. The aim of this work is the development of an iterative, gradient-based solution strategy, which can be applied even to complex multibody systems. The so-called adjoint method is a promising way to compute the direction of the steepest descent, i.e., the variation of a control signal causing the largest local decrease of the cost functional. The proposed approach will be more robust than solving the underlying boundary value problem, as the cost functional will be minimized iteratively while approaching the final conditions. Moreover, so-called influence differential equations are formulated to relate the changes of the controls and of the final conditions. In order to meet the end conditions, we introduce a descent direction that, on the one hand, approaches the optimum of the constrained cost functional and, on the other hand, reduces the error in the prescribed final conditions.

In this article, we discuss a special class of time-optimal control problems for dynamic systems, where the final state of a system lies on a hyper-surface. In time domain, this endpoint constraint may be given by a scalar equation, which we call transversality condition. It is well known that such problems can be transformed to a two-point boundary value problem, which is usually hard to solve, and requires an initial guess close to the optimal solution. Hence, we propose a new gradient-based iterative solution strategy instead, where the gradient of the cost functional, i.e., of the final time, is computed with the adjoint method. Two formulations of the adjoint method are presented in order to solve such control problems. First, we consider a hybrid approach, where the state equations and the adjoint equations are formulated in time domain but the controls and the gradient formula are transformed to a spatial variable with fixed boundaries. Second, we introduce an alternative approach, in which we carry out a complete elimination of the time coordinate and utilize a formulation in the space domain. Both approaches are robust with respect to poor initial controls and yield a shorter final time and, hence, an improved control after every iteration. The presented method is tested with two classical examples from satellite and vehicle dynamics. However, it can also be extended to more complex systems, which are used in industrial applications.

Cross laminated timber (CLT), as a structural plate-like timber product, has been established as a load bearing product for walls, floor and roof elements. In a bending situation due to the transverse shear flexibility of the crossing layers, the warping of the cross section follows a zigzag pattern which should be considered in the calculation model. The Refined Zigzag Theory (RZT) can fulfill this requirement in a very simple and efficient way. The RZT is a very robust and accurate analysis tool, which can handle the typical zigag warping of the cross section by introducing only one additional kinematic degree of freedom in case of plane beams and two more in case of biaxial bending of plates. Thus, the RZT-kinematics is able to reflect the specific and local stress behaviour near concentrated loads in combination with a warping constraint, while most other theories do not. The paper shows that the RZT is very well suited to analyze all kinds, of shear-elastic structural element like CLT-plate, timber-concrete composite structure or doweled beam in an accurate and unified way.

This article illustrates a novel approach for the determination of time-optimal controls for dynamic systems under observance of end conditions. Such problems arise in robotics, e.g. if the control of a robot has to be designed such that the time for a rest-to-rest maneuver becomes a minimum. So far, such problems have been considered as two-point boundary value problems, which are hard to solve and require an initial guess close to the optimal solution. The aim of this contribution is the development of an iterative, gradient based solution strategy for solving such problems. As an example, a Moon-landing as in the Apollo program, will be considered. In detail, we discuss the ascent, descent and abort maneuvers of the Apollo Lunar Excursion Module (LEM) to and from the Moon’s surface in minimum time. The goal is to find the control of the thrust nozzle of the LEM to minimize the final time.

In multibody dynamics, the Euler parameters are often used for the numerical simulation of rigid body rotations because they lead to a relatively simple form of the rotation matrix which avoids the evaluation of trigonometric functions and can thus save computational time. The Newmark method and the closely related Hilber–Hughes–Taylor (HHT) method are widely employed for solving the equations of motion of mechanical systems. They can also be applied to constrained systems described by differential algebraic equations. However, in the classical versions, the use of these integration schemes have a very unfavorable impact on the Euler parameter description of rotational motions. In this paper, we show analytically that the angular velocity for a rotation about a single axis under a constant moment will not increase linearly but grows slower. This effect, which does not appear for Euler angles, can be even observed if the numerical damping parameter α in the HHT method is set to zero. To circumvent this problem without losing the advantage of Euler parameters, we present a modified HHT method which reduces the damping effect on the angular velocity significantly and eliminates it completely for α=0.

The floating frame of reference formulation is an established method for the description of linear elastic bodies within multibody dynamics. An exact derivation leads to rather complex equations of motion. In order to reduce the computational burden, it is common to neglect certain terms. In the literature this is done by strict application of the small deformation assumption to the kinetic energy. This leads to a remarkably simplified set of equations. In this work, the significance of all terms is investigated at the level of the equations of motion. It is shown that for a large number of applications the previously mentioned set of simple equations is sufficient. Furthermore, scenarios are described in which this simple set is no longer accurate enough. Finally, guidelines are provided, so that engineers can decide which terms should be considered or not. The theoretical conclusions drawn in this work are underlined by qualitative numerical investigations.

During test rides of motorcycles modifications are made to the suspension. In order to quantify those changes, the nonlinear spring and damper characteristics must be determined. This is usually done on a test bench. However, measurements on a test bench are closely related to high costs and high time exposure. Hence, a parameter identification after a test run, formulated as an optimization task, seems to be an auspicious approach. For this purpose a cost function is defined, which is minimized by considering the dynamics of the system. The strength of the contribution is the efficient gradient computation using the adjoint variable approach. In order to approximate the nonlinear spring and damper characteristics cubic splines are used. The values of the spline functions at specified grid points (knots) are adjusted such that the deviation between simulation and measurement is minimal.

The Refined Zigzag Theory (RZT), developed by Tessler/Di Sciuva/Gherlone is among the most promising approaches for analyzing shear-elastic composite structures today. Since its appearance many contributions have been published dealing with finite elements for laminated structures based on the efficient kinematic of RZT. For composite beam-columns C0-elements of different orders as well as p-type approximations are formulated and assessed.
In this work a different approach is given. After establishing the governing equations in a first order differential equation system, the transfer matrix is obtained by a matrix series solution and by similarity transformation. The transfer matrix approach, in principle suited for 1D-structural elements such as beams, disks, circular plates and rotational shells, has been successfully applied in the past. Sometimes this approach exhibits numerical instabilities. The well-known relations between the transfer- and stiffness matrix are invoked to circumvent this drawback. The dynamic stiffness matrix and the load vector are obtained by reordering and partially inverting the submatrices of the transfer matrix. The results, which are obtained by one finite element only, are in agreement with available analytical and numerical solutions.

In order to achieve a correct representation of jointed structures within multibody dynamic simulations, an accurate computation of the nonlinear contact and friction forces between the contact surfaces is required. In recent history, trial vectors based on trial vector derivatives, the so-called joint modes, have been presented, which allow an accurate and efficient representation of this joint contact. In this paper, a systematic adaption of this method for preloaded bolted joints is presented. The new strategy leads to a lower number of additional joint modes required for accurate results and hence to lower computational time. Further, a major reduction of the computational effort for joint modes can be achieved. The potential and also possible limitations of the method are investigated using two numerical examples of a preloaded friction bar and a bolted piston rod bearing cap.

The adjoint method is an elegant approach for the computation of the gradient of a cost function to identify a set of parameters. An additional set of differential equations has to be solved to compute the adjoint variables, which are further used for the gradient computation. However, the accuracy of the numerical solution of the adjoint differential equation has a great impact on the gradient. Hence, an alternative approach is the discrete adjoint method, where the adjoint differential equations are replaced by algebraic equations. Therefore, a finite difference scheme is constructed for the adjoint system directly from the numerical time integration method. The method provides the exact gradient of the discretized cost function subjected to the discretized equations of motion.

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.

In this paper we deal with the problem of ill-conditioned reduced order models in the context of redundant formulated nonlinear multibody system dynamics. Proper Orthogonal Decomposition is applied to reduce the physical coordinates, resulting in an overdetermined system. As the original set of algebraic constraint equations becomes, at least partially, redundant, we propose a generalized constraint reduction method, based on the ideas of Principal Component Analysis, to identify a unique and well-conditioned set of reduced constraint equations. Finally, a combination of reduced physical coordinates and reduced constraint coordinates are applied to one purely rigid and one partly flexible large-scale model, pointing out method strengths but also applicability limitations.

This paper deals with the dynamics of jointed flexible structures in multibody simulations. Joints are areas where the surfaces of substructures come into contact, for example, screwed or bolted joints. Depending on the spatial distribution of the joint, the overall dynamic behavior can be influenced significantly. Therefore, it is essential to consider the nonlinear contact and friction phenomena over the entire joint. In multibody dynamics, flexible bodies are often treated by the use of reduction methods, such as component mode synthesis (CMS). For jointed flexible structures, it is important to accurately compute the local deformations inside the joint in order to get a realistic representation of the nonlinear contact and friction forces. CMS alone is not suitable for the capture of these local nonlinearities and therefore is extended in this paper with problem-oriented trial vectors. The computation of these trial vectors is based on trial vector derivatives of the CMS reduction base. This paper describes the application of this extended reduction method to general multibody systems, under consideration of the contact and friction forces in the vector of generalized forces and the Jacobian. To ensure accuracy and numerical efficiency, different contact and friction models are investigated and evaluated. The complete strategy is applied to a multibody system containing a multilayered flexible structure. The numerical results confirm that the method leads to accurate results with low computational effort.

The adjoint method is a very efficient way to compute the gradient of a cost functional associated to a dynamical system depending on a set of input signals. However, the numerical solution of the adjoint differential equations raises several questions with respect to stability and accuracy. An alternative and maybe more natural approach is the discrete adjoint method (DAM), which constructs a finite difference scheme for the adjoint system directly from the numerical solution procedure, which is used for the solution of the equations of motion. The method delivers the exact gradient of the discretized cost functional subjected to the discretized equations of motion. For the application of the discrete adjoint method to the forward solver, several matrices are necessary. In this contribution, the matrices are derived for the simple Euler explicit method and for the classical implicit Hilber–Hughes–Taylor (HHT) solver.

We present a method for optimizing inputs of multibody systems for a subsequently performed parameter identification. Herein, optimality with respect to identifiability is attained by maximizing the information content in measurements described by the Fisher information matrix. For solving the resulting optimization problem, the adjoint system of the sensitivity differential equation system is employed. The proposed approach combines these two well-established methods and can be applied to multibody systems in a systematic, automated manner. Furthermore, additional optimization goals can be added and used to find inputs satisfying, for example, end conditions or state constraints.

The present paper shows the embedding of the adjoint method in multibody dynamics and its broad applicability for examples for both, parameter identification and optimal control. Especially, in case of parameter identifications in engineering multibody applications, a theoretical enhancement of the proposed adjoint method by an error control of accelerations is inevitable in order to meet the circumstances of experimental studies using acceleration sensors in general.

Many models of three-dimensional rigid body dynamics employ Euler parameters as rotational coordinates. Since the four Euler parameters are not independent, one has to consider the quaternion constraint in the equations of motion. This is usually done by the Lagrange multiplier technique. In the present paper, various forms of the rotational equations of motion will be derived, and it will be shown that they can be transformed into each other. Special attention is hereby given to the value of the Lagrange multiplier and the complexity of terms representing the inertia forces. Particular attention is also paid to the rotational generalized external force vector, which is not unique when using Euler parameters as rotational coordinates.

The present paper illustrates the potential of the adjoint method for a wide range of optimization problems in multibody dynamics such as inverse dynamics and parameter identification.
Although the equations and matrices included show a complicated structure, the additional effort when combining the standard forward solver to the adjoint backward solver, is kept in limits.
Therefore, the adjoint method shows an efficient way to incorporate inverse dynamics to engineering multibody applications, e.g., trajectory tracking or parameter identification in the field of robotics.
The present paper studies examples for both, parameter identification and optimal control, and shows the potential of the adjoint method in solving classical optimization problems in multibody dynamics.

In case of complex multibody systems, an efficient and time-saving computation of the equations of motion is essential, in particular concerning the inertia forces. When using the floating frame of reference formulation for modeling a multibody system, the inertia forces, which include velocity-dependent forces, depend non-linearly on the system state and therefore have to be updated in each time step of the dynamic simulation. Since the emphasis of the present investigation is on the efficient computation of the velocity-dependent inertia forces as well as a fast simulation of multibody systems, a detailed derivation of the latter forces for the case of a general rotational parametrization is given. It has to be emphasized that the present investigations revealed a simpler representation of the velocity-dependent inertia forces compared to results presented in literature. In contrast to the formulas presented in literature, the presented formulas do not depend on the type of utilized rotational parametrization or on any associated assumptions.

The focus of the present article lies on new enhanced beam finite element formulations in the absolute nodal coordinate formulation (ANCF) and its embedding in the available formulations in the literature. The ANCF has been developed in the past for the modeling of large deformations in multibody dynamics problems. In contrast to the classical nonlinear beam finite elements in literature, the ANCF does not use rotational degrees of freedom, but slope vectors for the parameterization of the orientation of the cross section. This leads to several advantages compared to the classical formulations, e.g. ANCF elements do not necessarily suffer from singularities emerging from the parameterization of rotations. In the classical large rotation vector formulation, the mass matrix is not constant with respect to the generalized coordinates. In the case of ANCF elements, a constant mass matrix follows, which is advantageous in dynamic analysis.
Within the present paper the state of the art of ANCF elements in literature is reviewed including the basic description of the kinematics, interpolation strategies and definition of elastic forces, but also problems and known disadvantages arising in the existing elements, as e.g. the locking phenomenon.
It has to be mentioned here that most of the studies on nonlinear elements based on the ANCF in literature use linear constitutive laws. Regarding many applications in which geometric and material nonlinearities arise, elastic material models are not sufficient to represent the real problem accurately. For this reason, an extension of the material model is necessary in order to fulfill the requirements of current but also of future materials arising in engineering or research. An overview of existing nonlinear material models in literature can be found in the “Appendix”.

The dynamic simulation of contact in rolling processes is very time-consuming. This is mainly based on the fine resolution of the surface domain of each roll, which, however, is essential in order to capture the effect of concentrated contact forces. Existing model order reduction techniques cannot be readily applied due to the nonlinear nature of the contact dynamics. In order to improve the speed of contact analysis, the present paper proposes a sophisticated combination of so-called characteristic static correction modes and vibration normal modes for describing the deformation of each roll. While the characteristic static correction modes are required to capture the concentrated nonlinear contact forces, the vibration normal modes describe the global deformation behavior of the rolls. For the computation of the characteristic static correction modes, first attachment modes are computed for a longitudinal sub-area of each roll. Then an eigenanalysis is performed on the component mode synthesis mass and stiffness matrices that correspond to the attachment modes. The resultant eigenvectors have been truncated and applied to the entire surface domain of the rolls. In order to obtain well-conditioned equations, all modes are finally orthonormalized. An example from the metal forming industry is used to demonstrate the results.

The present paper introduces the non-commercial academic MBS-software FREEDYN, which is currently developed at the University of Applied Sciences Upper Austria. Special attention is turned on the efficient computation of highly nonlinear multibody dynamics systems including flexible components arising in nearly every industrial brunch. A fast and accurate simulation code and its open-source release are the main goals. Several examples have been performed in order to show its user-friendly applicability and its accuracy and efficiency according to the analytically derived components of the Jacobian matrix as well as their description using inertia shape integrals.

This paper considers solution strategies for “dynamical inverse problems”, where the main goal is to determine the excitation of a dynamical system, such that some output variables, which are derived from the system’s state variables, coincide with desired time functions. The paper demonstrates how such problems can be restated as optimal control problems and presents a numerical solution approach based on the method of steepest descent. First, a Performance measure is introduced, which characterizes the deviation of the output variables from the desired values and which is minimized by the solution of the inverse problem. Second, we show, how the gradient of this error functional can be computed efficiently by applying the theory of optimal control, in particular by following an idea of H. J. Kelley and A. E. Bryson. As the major contribution of this paper we present a modification of this method which allows the application to the case where the state equations are given by a set of differential algebraic equations. This situation has great practical importance since multibody systems are mostly described in this way. For comparison we also discuss an approach which bases an a direct transcription of the optimal control problem. Moreover, other methods to solve dynamical inverse problems are summarized.