Large Rigid Body Rotations Research Articles

Optimal selection of suspension parameters in large scale vehicle models

Optimum values are selected for the suspension damping and stiffness parameters of complex car models, subjected to road excitation, by applying suitable numerical methodologies. These models result from a detailed finite-element discretisation and possess a relatively large number of degrees of freedom. They also involve strongly nonlinear characteristics, due mostly to large rigid body rotation of some of their components and the properties of the connection elements. First, attention is focused on gaining some insight into the dynamics of the mechanical models examined, resulting when the vehicle passes over roads involving typical geometric profiles. Then, the emphasis is shifted to presenting results obtained by applying appropriate optimisation methodologies. For this purpose, three classes of design criteria are first set up, referring to passenger ride comfort, suspension travel and car road holding and yielding the most important suspension stiffness and damping parameters. Originally, the optimisation is performed by forming a composite cost function and employing a single-objective optimisation method. Since the design criteria are conflicting, a multi-objective optimisation methodology is also set up and applied subsequently.

Dynamic modeling of cable system using a new nodal position finite element method

AbstractThe handling of cable systems onboard surface ships and submarines presents a significant technical challenge to design engineers in defense and ocean industries. The current approaches rely heavily on empirical methods and time‐consuming/expensive prototype testing. Computer simulation provides a cost effective way to reduce the high risks associated with the design of the cable‐towed system. This paper presents a new nodal position finite element method (FEM) to effectively simulate the dynamics of cable system experiencing large rigid body rotation coupled with small elastic deformation. The existing FEM deals with the large rigid body rotation by linearized incremental approximation that is prone to numerical inaccuracies resulting from large 3D rotations. By solving for the position directly instead of indirectly via the displacement, the new FEM does not need to decouple the elastic deformation from the rigid body rotation and thus eliminates the error source arising from the linearized incremental approximation in existing FEM. Analysis results demonstrate that the new FEM algorithm is simple and robust by comparing with numerical benchmark test and experiments including sea trial data. Copyright © 2008 John Wiley & Sons, Ltd.

Nonlinear Flutter Calculations Using Finite Elements in a Direct Eulerian-Lagrangian Formulation

A fully nonlinear aeroelastic formulation of the direct Eulerian-Lagrangian computational scheme is presented in which both structural and aerodynamic nonlinearities are treated without approximations. The method is direct in the sense that the calculations are done at the finite element level, both in the fluid and structural domains, and the fluid-structure system is time-marched as a single dynamic system using a multistage Runge-Kutta scheme. The exact nonlinear boundary condition at the fluid-structure boundary is satisfied based on the actual deformation of the wing. The generalized forces associated with the in-plane and out-of-plane degrees of freedom are calculated in local Lagrangian element coordinate systems that fully account for large rigid-body translations and rotations. Finite rotation relations are used to update the nodal deformation vectors at the end of each time step. Numerical results are presented for several nonlinear static and dynamic examples for which published results are available. Results of aeroelastic calculations using the new nonlinear model demonstrate the importance of including the nonlinear stiffening arising from the in-plane strains when calculating limit-cycle-oscillation amplitudes of wings of low-to-moderate aspect ratios and the limitations of the von Karman nonlinear plate model in these cases.

Modelling and ride dynamics of a flexible multi-body model of an urban bus

Dynamic response of a large order mechanical model of an urban bus is investigated in this paper. The emphasis is first put on developing a quite complete model, which can be utilized in order to extract sufficiently reliable and accurate information related to its dynamics in a fast way. Since some of the components of the bus undergo large rigid body rotation, in addition to motion resulting from their deformability, a multi-body dynamics framework is adopted. This implies that the resulting equations of motion appear in the form of a strongly non-linear set of differential-algebraic equations, which are difficult to handle even numerically. In fact, the modelling becomes more involved because all the significant non-linearities appearing in the interconnections of the structural components and especially in the front and rear suspension subsystems of the bus are taken into account. In order to alleviate some of these complexities, the number of degrees of freedom of each component, associated with its deformability, is reduced drastically by applying an appropriate coordinate condensation methodology. Finally, this model is employed and numerical results are obtained for motions resulting from typical road excitation. In particular, selected response quantities related to ride comfort are examined for characteristic combinations of the bus suspension stiffness and damping parameters.

Enhanced assumed strain (EAS) and assumed natural strain (ANS) methods for one‐point quadrature solid‐shell elements

AbstractA reduced enhanced solid‐shell (RESS) finite element concept has been suggested recently by Alves de Sousa et al. (Int. J. Numer. Meth. Engng 2005; 62:952; Int. J. Numer. Meth. Engng 2006; 67:160; Int. J. Plasticity 2007; 23:490). Developments on the ‘RESS’ element were motivated by the following reasons: first, solid‐shell elements automatically incorporate the normal stress along the thickness direction, which makes them more suitable for the simulations with double‐sided contact than their shell counterparts; second, they have only translational degrees of freedom, which alleviates some difficulties associated with formulating complex shell formulations using nodal rotations; third, general constitutive models can be used and a reformulation for plane‐stress conditions is not necessary. Furthermore, traditionally, solid elements have been developed based on fully integrated schemes with limited number of integration points per single element layers which renders them computationally expensive. On the contrary, the solid‐shell element by Alves de Sousa et al. (Int. J. Numer. Meth. Engng 2005; 62:952; Int. J. Numer. Meth. Engng 2006; 67:160; Int. J. Plasticity 2007; 23:490) was developed for one single‐layer shell structure with reduced in‐plane and multiple integration points along the thickness direction of the shell. The formulation consists of several combinations of well‐known techniques to ameliorate locking problems as is the case of the enhanced assumed strain (EAS) method. However, the proposed ‘RESS’ solid shell did not consider the transverse shear components of hourglass (or physical stabilization) parts. This negligence produces a slightly flexible behavior of the element, but it can also cause the appearance of hourglass modes for several non‐linear applications including large rigid body rotations. Sometimes, the negligence brings a non‐positive‐definite status on the stiffness matrix. In order to overcome such drawbacks, a modified assumed natural strain (ANS) method considering the top and bottom surfaces of the element was incorporated for the transverse shear components. At the same time, new hourglass strains for the membrane field were constructed based on the stabilization vectors of Liu et al. (Comput. Meth. Appl. Mech. Engng 1998; 154:69). With these modifications, the improved element (called ‘M‐RESS’) passes both the membrane and bending patch tests and performs with remarkable stability and accuracy in sheet‐forming simulations. Copyright © 2007 John Wiley & Sons, Ltd.

Global and local coordinates in digital image correlation

Digital image correlation (DIC) is commonly used to measure specimen displacements by correlating an image of a specimen in an undeformed or reference configuration and a second image under load. To establish the correlation between the images, numerical techniques are used to locate an initially square image subset in a reference image. In this process, choosing appropriate coordinates is of fundamental importance to ensure accurate results. Both global and local coordinates can be used in shape functions. However, large rigid body rotations and deformations are accurately obtained by using global rather than local shape functions. In addition, points located after displacement may not be at an integer pixel distance from the original position. Hence subpixel displacement estimation methods such as interpolation or fitting of correlation coefficients are essential. A solution using the least-squares method is employed by choosing proper coordinates, and the feasibility of using local coordinates is demonstrated and validated with a mathematical model. Both simulated and experimental results show that the proper choice of coordinates does ensure the reliability and improve the accuracy of measurements in DIC.

The Absolute Coordinate Formulation with Elasto-Plastic Deformations

The present work contributes to the field of multibody systems with respect to the absolute coordinate formulation with a reduced expression of the strain energy and a non-linear constitutive model. Standard methods for multibody systems lead to highly non-linear terms either in the mass matrix or in the stiffness matrix and the most expensive part in the solution of the equations of motion is the assembling of these matrices, the computation of the Jacobian of the non-linear system and the solution of a linear system with the system matrices. In the present work, a consistent simplification of the equations of motion with respect to small deformations but large rigid-body motions is performed. The absolute coordinate formulation is used, therefore the total displacements are the unknowns. This formulation leads to a constant mass matrix while the non-linear stiffness matrix is composed of the constant small strain stiffness matrix rotated by the underlying rigid body rotation. Plastic strains are introduced by an additive split of the strain into an elastic and a plastic part, a yield condition and an associative flow rule. The decomposition of strain has to be performed carefully in order to obey the principle of objectivity for plasticity under large rigid body rotations. As an example, a two-dimensional plate which is hinged at one side and driven by a harmonic force at the opposite side is considered. Plastic deformation is assumed to occur due to extreme environmental influences or due to failure of some attached parts like a defect bearing.

Dynamics of Slider-Crank Mechanisms with Flexible Supports and Non-Ideal Forcing

Transient and steady state dynamic response of a class of slider-crank mechanisms is investigated. Specifically, the class of mechanisms examined involves rigid members but compliant supporting bearings. Moreover, the mechanisms are subjected to non-ideal forcing. Namely, both the driving and the resisting loads are expressed as a function of the angular coordinate describing the crank rotation. First, an appropriate set of equations of motion is derived by applying Lagrange's equations. These equations are strongly nonlinear due to the large rigid body rotation of the crank and the connecting rod, as well as due to the nonlinearities associated with the bearing action and the form of the driving and the resisting loads. Consequently, the dynamics of the resulting dynamical system is examined by solving the equations of motion numerically. More specifically, transient response is captured by direct integration, while determination of complete branches of steady state response is achieved by applying appropriate numerical methodologies. Initially, mechanisms whose crankshaft is supported by bearings with rolling elements and linear stiffness characteristics are examined. Then, numerical results are presented for rolling element bearings with nonlinear stiffness characteristics. Finally, the study is focused on mechanisms supported by hydrodynamic bearings. In all cases, the attention is focused on investigating the influence of the system parameters on its dynamics. Moreover, models with constant crank angular velocity are first analysed, since they provide valuable insight into some aspects of the system dynamics. Eventually, the emphasis is shifted to the general case of non-ideal forcing, originating from the dependence of the driving and the resisting moments on the crankshaft motion.

Use of the Finite Element Absolute Nodal Coordinate Formulation in Modeling Slope Discontinuity

A large rigid body rotation of a finite element can be described by rotating the axes of the element coordinate system or by keeping the axes unchanged and change the slopes or the position vector gradients. In the first method, the definition of the local element parameters (spatial coordinates) changes with respect to a body or a global coordinate system. The use of this method will always lead to a nonlinear mass matrix and non-zero centrifugal and Coriolis forces. The second method, in which the axes of the element coordinate system do not rotate with respect to the body or the global coordinate system, leads to a constant mass matrix and zero centrifugal and Coriolis forces when the absolute nodal coordinate formulation is used. This important property remains in effect even in the case of flexible bodies with slope discontinuities. The concept employed to accomplish this goal resembles the concept of the intermediate element coordinate system previously adopted in the finite element floating frame of reference formulation. It is shown in this paper that the absolute nodal coordinate formulation that leads to exact representation of the rigid body dynamics can be effectively used in the analysis of complex structures with slope discontinuities. The analysis presented in this paper also demonstrates that objectivity is not an issue when the absolute nodal coordinate formulation is used due to the fact that this formulation automatically accounts for the proper coordinate transformations.

Topology optimization of 2D‐frame structures with path‐dependent response

AbstractThis paper deals with topology stiffness optimization of a ground structure consisting of rectangular 2D‐beam elements with path‐dependent response. The objective of the optimization is to minimize the compliance of the structure when the mass is prescribed. The path‐dependent response of the structure is due to the plastic hinges at the ends of the beam elements. The plastic hinges are modelled through yield surfaces giving the interaction between the stress resultants when the cross‐section is fully plastified. In combination with the yield surface a simple hardening rule is applied. The geometrical non‐linearities are also included, so the beam elements can perform large rigid body rotations. The finite element solution is obtained with an implicit backward Euler algorithm. From the finite element solution the analytical sensitivities are computed using the direct method. Copyright © 2003 John Wiley & Sons, Ltd.

剛体回転に関する不連続変形法 (DDA) の研究

The separation of rigid body rotation term, r0, from the other displacement variables is a breakthrough idea in original discontinuous deformation analysis (DDA) formula. Although the original linear displacement function is a fast and efficient method in computation, it loses accuracy when blocks undergo large rigid body rotations. It can be a significant problem especially when solving the problem with high speed rigid body rotation, such as rock fall problem, or the larger time interval is introduced to the problem with rigid body rotation to obtain the results within less computation steps. Up to date, it is only known that the large rigid body rotation can cause free expansion and change the weight of block in the analysis. However, the authors consider that the problem can also affect the contact judgments in open-close iterations and cause wrong contact forces in the computation. In this paper, this problem will be discussed, and a new method named “post contact adjustment” is developed and applied to the contact computation terms of DDA. After the improvements, the simulation results show better contact computations and block area preservations even when the large rigid body rotation is carried out.

Nonlinear response of shell structures: effects of plasticity modelling and large rotations

Many structural applications require nonlinear finite element analyses in order to assess response and capacity. Plastic deformations may be accounted for by means of thickness integration or stress resultants. The stress resultant model employed herein is based on Ilyushins' linear yield criterion for thin shells. The corners present with this criterion are circumvented by means of a simplification, hence, there is no need for multi-surface stress resultant updates. A backward Euler difference is employed in the stress resultant update, and a consistent tangent is used in the Newton–Raphson iterations on the global equilibrium. Limit points are traversed by means of an orthogonal trajectory method. The response of compression dominated shells with imperfections typically corresponds to limit point behaviour. For stress resultant plasticity, the nonlinear transition from initial yield to full plasticity in shell bending is missed. Hence, the efficiency obtained by eliminating thickness integration is countered by some inaccuracy in the response simulation. This is investigated by means of comparison with finite element simulations employing integration through thickness (with linear or nonlinear hardening). Both steel and aluminium alloys are considered. In collapse response of slender structures, the straining of the material may be moderate, but the motion may be governed by large rigid body translations and rotations. A way of accounting for this by means of the co-rotated approach is presented. Triangular high-performance facet shell elements are employed. By example computations, the importance of nonlinear geometry contributions is illustrated.

“Open” structures of MurD: domain movements and structural similarities with folylpolyglutamate synthetase

UDP- N-acetylmuramoyl- l-alanine: d-glutamate (MurD) ligase catalyses the addition of d-glutamate to the nucleotide precursor UDP- N-acetylmuramoyl- l-alanine (UMA). The crystal structures of Escherichia coli in the substrate-free form and MurD complexed with UMA have been determined at 2.4 Å and 1.88 Å resolution, respectively. The MurD structure comprises three domains each of a topology reminiscent of nucleotide-binding folds. In the two structures the C-terminal domain undergoes a large rigid-body rotation away from the N-terminal and central domains. These two “open” structures were compared with the four published “closed” structures of MurD. In addition the comparison reveals which regions are affected by the binding of UMA, ATP and d-Glu. Also we compare and discuss two structurally characterized enzymes which belong to the same ligase superfamily: MurD and folylpolyglutamate synthetase (FGS). The analysis allows the identification of key residues involved in the reaction mechanism of FGS. The determination of the two “open” conformation structures represents a new step towards the complete elucidation of the enzymatic mechanism of the MurD ligase.

Collapse of thin shell structures?stress resultant plasticity modelling within a co-rotated ANDES finite element formulation

Due to the very non-linear behaviour of thin shells under collapse, numerical simulations are subject to challenges. Shell finite elements are attractive in these simulations. Rotational degrees of freedom do, however, complicate the solution. In the present study a co-rotated formulation is employed. The deformation of the shell is decomposed in to a contribution from large rigid body rotation and a strain producing term. A triangular assumed strain shell finite element is used. Hence, a high performance elastic element is combined with the co-rotated formulation. In the co-rotated co-ordinate system the plasticity is accounted for by a simplifyed Ilyushin stress resultant yield surface. The stress update is determined from the backward Euler difference, and a consistent geometrical and material tangent stiffness is derived. Comparison with other published analysis results show that the present formulation gives acceptable accuracy. Copyright © 1999 John Wiley & Sons, Ltd.

Collapse of thin shell structures—stress resultant plasticity modelling within a co‐rotated ANDES finite element formulation

Due to the very non-linear behaviour of thin shells under collapse, numerical simulations are subject to challenges. Shell finite elements are attractive in these simulations. Rotational degrees of freedom do, however, complicate the solution. In the present study a co-rotated formulation is employed. The deformation of the shell is decomposed in to a contribution from large rigid body rotation and a strain producing term. A triangular assumed strain shell finite element is used. Hence, a high performance elastic element is combined with the co-rotated formulation. In the co-rotated co-ordinate system the plasticity is accounted for by a simplifyed Ilyushin stress resultant yield surface. The stress update is determined from the backward Euler difference, and a consistent geometrical and material tangent stiffness is derived. Comparison with other published analysis results show that the present formulation gives acceptable accuracy. Copyright © 1999 John Wiley & Sons, Ltd.

Modification of the DDA method for rigid block Problems

Abstract This paper presents the development of the rigid block version of discontinuous deformation analysis and its preliminary application in rockfall simulation. Modifications, including the reduction of the error due to large rigid body rotation by using post-correction at each time step of calculation and the incorporation of mass proportional damping to account for energy losses, are made to meet the requirement for large block translation as well as rotation. To improve the program efficiency, rigid body displacement function and preconditioned conjugate gradient solver are adopted. The results obtained show that the newly developed code, RIG-DDA, can appropriately and efficiently model various modes of rockfall movement and predict the trajectory of falling rock debris, which is helpful in the design of protection measures.

Structure of apo duck ovotransferrin: the structures of the N and C lobes are in the open form.

The structure of apo duck ovotransferrin (APODOT) has been determined at a resolution of 4.0 A by the molecular replacement method using the structure of duck ovotransferrin (DOT) as the search model. The DOT structure contains two iron binding sites; one in the N-terminal lobe lying between domains N1 and N2 and one in the C-terminal lobe between domains C1 and C2. Both lobes have a closed structure. Models of various forms of both the N and C lobes were used in the search. The final model was refined to give an R factor of 0.22. The comparison of the structure of APODOT with that of DOT shows that both the N and the C lobes are in an open form, where the N2 and C2 domains undergo large rigid-body rotations of 51.6 and 49.9 degrees relative to the N1 and C1 domains, respectively. The interface between the N and C lobes, which is formed by the N1-C1 contact in the core of the molecule does not change significantly. The DOT molecule may be described in terms of three rigid bodies; the N1 and C1 domains as one rigid body forming the static core of the molecule and the N2 and C2 domains as two other rigid bodies which, on the release of iron, move away from the static core of the molecule to form the open structure of APODOT.

Numerical integration of rate constitutive equations under finite deformation

An objective stress rate must be used in the constitutive law to accurately describe material response when large rigid body translations and rotations are present. The choice of a suitable objective stress rate has been the subject of considerable debate. For example, a basic flaw in the well known Jaumann stress rate (J-rate) has been demonstrated using the simple shear problem. A number of alternative objective stress rates have been suggested and developed. Among these is the Eulerian or E-rate in which rate constitutive calculations are performed on material axes coincident with the principal axes. The use of the principal axes technique greatly simplifies the constitutive calculations and resolves the difficulties surrounding the question of objectivity and frame invariance which arise in finite deformation problems. A numerical algorithm is presented which incorporates the principal axes technique for the Eulerian stress rate. The algorithm is intended for use in general purpose two or three dimensional solid mechanics software using the incremental or rate form of constitutive modelling. The algorithm focuses on an efficient method of integrating the material rotation tensor relating the principal axes to the fixed or global axes. The implementation of this algorithm into a general purpose non-linear transient dynamic finite element computer code is included. A number of numerical examples ranging from the simple shear problem to a large scale impact simulation demonstrates the accuracy and salient features of the algorithm.

Calculation of non-linear vibration of rotating beams by using tetrahedral and solid finite elements

A development is presented of the non-linear dynamic equations that govern the motion of the tetrahedral and solid finite elements that undergo large displacements. The development presented is exemplified by using the four-node, 12-degree-of-freedom tetrahedral element and the eight-node, 24-degree-of-freedom solid element. It is shown that the element shape functions used in this investigation can be used to describe large translations and finite rigid body rotations. Accordingly, the non-linear formulation presented in this paper can be used in the analysis of small as well as large deformation. The element configuration is identified by using four co-ordinate systems. These co-ordinate systems are the global, body, element and intermediate element co-ordinate systems. The large displacement of the tetrahedral and solid elements is described by using a set of absolute co-ordinates that define the location and orientation of the deformable body co-ordinate systems. The non-linear differential equations of motion of the tetrahedral and solid finite elements are developed by using the principle of virtual work in dynamics. The use of the non-linear dynamic formulation presented in this investigation is demonstrated by using a flexible single robotic arm manipulator that undergoes large displacements. The results obtained by using the four-node tetrahedral elements and the eight-node solid elements are compared with the results obtained by using the three-dimensional beam element. This comparison shows that the discrepancy between the results obtained by using the solid and tetrahedral elements in beam problems is more significant in the dynamic analysis as compared to the discrepancy of 10% reported in the literature for the static analysis.

Assembly of Finite‐Element Helicopter Subsystems with Large Relative Rotations

A general finite‐element procedure is presented for modeling rotorcraft undergoing elastic deformations in addition to large rigid body motions with respect to inertial space. Special attention is given to the coupling of the rotor and fuselage subsystems subject to large relative rotations. Initially, the rotor and fuselage subsystems are assembled separately as small‐rotation finite‐element models in a moving coordinate system. In order to handle large rigid body rotations, the coordinate systems are tied to the structure using one of several alternate constraint methods. Finally, the equations which allow large rotations are constrained together using a rotating to nonrotating transformation which allows rotor azimuth angle as a degree of freedom. The resulting system of equations, which has not been implemented, is applicable to both helicopter trim and large angle maneuver analyses.