A nonsmooth frictional contact formulation for multibody system dynamics

We present a new node-to-face frictional contact element for the simulation of the nonsmooth dynamics of systems composed of rigid and flexible bodies connected by kinematic joints. The equations of motion are integrated using a nonsmooth generalized-α time integration scheme and the frictional contact problem is formulated using a mixed approach, based on an augmented Lagrangian technique and a Coulomb friction law. The numerical results are independent of any user-defined penalty parameter for the normal or tangential component of the forces and, the bilateral and the unilateral constraints are exactly fulfilled both at position and velocity levels. Finally, the robustness and the performance of the proposed algorithm are demonstrated by solving several numerical examples of nonsmooth mechanical systems involving frictional contact.     

A generalized recursive formulation for constrained flexible multibody dynamics

This research develops a relative co‐ordinate formulation for the multibody flexible dynamics. The velocity transformation method is notationally compact, because the Cartesian generalized velocities are simultaneously transformed to the relative generalized velocities in a matrix form. However, inherent computational efficiency in the recursive kinematics between two adjacent bodies has not been exploited. This research presents a recursive formulation which is both notationally compact and computationally efficient. The velocity transformation method is used to derive the equations of motion and their derivatives. Matrix operations associated with the velocity transformation matrix in the resulting equations of motion and their derivatives are classified into several categories. A joint library of the generalized recursive formulas is developed for each category. When one category is encountered in implementing the equations of motion and their derivatives, the corresponding recursive formulas in the category are invoked. When a new force or joint module is added to a general purpose programme in the relative co‐ordinate formulation, the modules for the rigid body are not reusable for the flexible body. Since the flexible body dynamics handles additional generalized co‐ordinates associated with deformation, implementation of the flexible dynamics is generally complicated and prone to coding mistakes. A virtual rigid body is introduced at every joint and force reference frames. A virtual flexible body joint is introduced between two body reference frames of the virtual and original bodies. This makes a flexible body subjected to only the kinematic admissibility condition for the virtual flexible body joint. As a result, the only extra work to handle the flexible bodies is to add the virtual flexible body joint modules in all recursive formulas. Since computation time in a relative co‐ordinate formulation is approximately proportional to the number of relative co‐ordinates, computational overhead due to the additional virtual bodies and joints are minor. Meanwhile, implementation convenience is dramatically improved. Copyright © 2001 John Wiley & Sons, Ltd.     

A general formulation in rigid multibody dynamics

The dynamics of rigid multibodies is traditionally formulated by means of either minimal or redundant co-ordinates methods. An alternative approach is here proposed whereby a highly redundant set of coordinates is adopted. As a result, the equations of motion of the constrained bodies are decoupled. Several meaningful parameters are directly available and the constraint conditions are enforced in a very natural way. The first part of the paper presents the basic meanings and the theoretical developments of the formulation. The second develops a numerical approximation for the methodology proposed in the first part. The non-linear system of differential-algebraic equations governing the motion of the multibody is reduced to its weak form. It is linearized by applying a Newton–Raphson procedure and approximated through the method of finite elements in time. The details of the numerical application of this method are discussed and a solution procedure is presented. Finally, some numerical examples involving tree and closed loop topologies prove the capability of the present formulation in handling multibody dynamics.     

Object-oriented implementation of a graph-theoretic formulation for planar multibody dynamics

The objective of this paper is to describe the object-oriented implementation and computational efficiency of a multibody dynamics algorithm for planar mechanical systems. The underlying formulation uses a unique combination of orthogonal projection techniques and graph-theoretic methods to automatically generate the equations of motion in terms of ‘branch’ co-ordinates that are selected by a user. The direct analogy between the physical components in a multibody system and the ‘objects’ in Object-Oriented Programming (OOP) methods is exploited in a C++ implementation of the dynamic formulation. Issues associated with this OOP implementation are discussed, and the results of computer simulations are presented and examined for different sets of user-selected co-ordinates. © 1997 John Wiley & Sons, Ltd.     

A GPU‐based preconditioned Newton‐Krylov solver for flexible multibody dynamics

This paper describes an approach to numerically approximate the time evolution of multibody systems with flexible (compliant) components. Its salient attribute is that at each time step, both the formulation of the system equations of motion and their numerical solution are carried out using parallel computing on graphics processing unit cards. The equations of motion are obtained using the absolute nodal coordinate formulation, yet any other multibody dynamics formalism would fit equally well the overall solution strategy outlined herein. The implicit numerical integration method adopted relies on a Newton–Krylov methodology and a parallel direct sparse solver to precondition the underlying linear system. The proposed approach, implemented in a software infrastructure available under an open‐source BSD‐3 license, leads to improvements in overall simulation times of up to one order of magnitude when compared with matrix‐free parallel solution approaches that do not use preconditioning. Copyright © 2015 John Wiley & Sons, Ltd.     

An accelerated iterative method for the dynamics of constrained multibody systems

An accelerated iterative method is suggested for the dynamic analysis of multibody systems consisting of interconnected rigid bodies. The Lagrange multipliers associated with the kinematic constraints are iteratively computed by the monotone reduction of the constraint error vector, and the resulting equations of motion are easily time-integrated by a well established ODE technique. The velocity and acceleration constraints as well as the position constraints are made to be satisfied at the joints at each time step. Exact solution is obtained without the time demanding procedures such as selection of the independent coordinates, decomposition of the constraint Jacobian matrix, and Newton Raphson iterations. An acceleration technique is employed for the faster convergence of the iterative scheme and the convergence analysis of the proposed iterative method is presented. Numerical solutions for the verification problems are presented to demonstrate the efficiency and accuracy of the suggested technique.     

A hybrid variational method for multibody dynamics

This paper presents a hybrid variational method to minimize computational effort in forming and solving the equations of motion for broad classes of rigid multibody mechanical systems. The hybrid method combines the O(n) and O(n³) recursive variational methods for forming the equations of motion in terms of joint relative co-ordinates. While the O(n³) method is more efficient than the O(n) method for systems with short chains and decoupled loops, the converse is true when the number of bodies in chains is large. The computational complexity of the O(n³) and O(n) methods in forming and solving the equations of motion is analysed as a function of the numbers of bodies, decoupled loops, joints, cut joints, cut-joint constraint equations and force elements. Based on complexity estimates, the method presented in this paper uses either the O(n) or O(n³) variational method to formulate the equations of motion for each open chain and decoupled loop in the system, to minimize the computational effort.     

An effective finite rotation formulation for geometrical non-linear shell structures

This paper presents a simple stress resultant 4-node shell element for geometrical non-linear analysis. In order to model smooth surfaces and/or stiffened structures, a simple and efficient technique for finite rotation is adopted. By means of suppressing the component of singular rotation effectivley, convenient use of six degrees of freedom is possible without deteriorating the robustness and the convergence rate of the classical 5-dof formulation. In the formulation of shell element, section eccentricity is also considered to model stiffened structures. Through numerical experiments the effectiveness of the proposed method is demonstrated. Received 16 November 2000     

Comparison of solution strategies for multibody dynamics equations

In the last decades, several different formalisms have been proposed in the literature in order to simulate the dynamics of multibody systems. First, the choice of the coordinates leads to very different formalisms. The set of constraint equations appears very different in complexity or in size depending on whether either relative, natural or reference point Cartesian coordinates are adopted. Second, once such a choice has been made, different solution strategies can be followed. It seems that formalisms based on redundant coordinates are often used in commercial software, whereas those based on a minimum number of coordinates are usually preferred in real‐time computations. In this paper, with reference to models with redundant absolute coordinates, the problem of coordinate reduction will be discussed and a group of 11 different methods will be compared on the basis of their computational efficiency. In particular, methods based on constraint orthogonalization and on the use of pseudoinverse matrices have been selected, together with other methods based on least‐squares block solution. The methods have been implemented on three different test cases. The main purpose is to provide hints and guidelines on the choice and availability of solution strategies during simulation of moderate size multibody systems. Copyright © 2011 John Wiley & Sons, Ltd.     

Scalable total BETI based solver for 3D multibody frictionless contact problems in mechanical engineering

A total BETI (TBETI) based domain decomposition algorithm with the preconditioning by a natural coarse grid of the rigid body motions is adapted for the solution of multibody frictionless contact problems of linear elastostatics and proved to be scalable, i.e., the cost of the solution is asymptotically proportional to the number of variables. The analysis admits floating bodies. The proofs combine the original results by Langer and Steinbach on the scalability of BETI for linear problems and our development of optimal quadratic programming algorithms for bound and equality constrained problems. The theoretical results are verified by numerical experiments. The power of the method is demonstrated on the analysis of ball bearings.     

An o(n) complexity recursive algorithm for multi‐flexible‐body dynamics based on absolute nodal coordinate formulation

A new algorithm called recursive absolute nodal coordinate formulation algorithm (REC‐ANCF) is presented for dynamic analysis of multi‐flexible‐body system including nonlinear large deformation. This method utilizes the absolute nodal coordinate formulation (ANCF) to describe flexible bodies, and establishes a kinematic and dynamic recursive relationship for the whole system based on the articulated‐body algorithm (ABA). In the ordinary differential equations (ODEs) obtained by the REC‐ANCF, a simple form of the system generalized Jacobian matrix and generalized mass matrix is obtained. Thus, a recursive forward dynamic solution is proposed to solve the ODEs one element by one element through an appropriate matrix manipulation. Utilizing the parent array to describe the topological structure, the REC‐ANCF is suitable for generalized tree multibody systems. Besides, the cutting joint method is used in simple closed‐loop systems to make sure the O(n) algorithm complexity of the REC‐ANCF. Compared with common ANCF algorithms, the REC‐ANCF has several advantages: the optimal algorithm complexity (O(n)) under limited processors, simple derivational process, no location or speed constraint violation problem, higher algorithm accuracy. The validity and efficiency of this method are verified by several numerical tests. Copyright © 2016 John Wiley & Sons, Ltd.     

A ‘nodeless’ dual superelement formulation for structural and multibody dynamics application to reduction of contact problems

A ‘nodeless’ superelement formulation based on dual‐component mode synthesis is proposed, in which the superelement dynamic behavior is described in terms of modal intensities playing the role of intrinsic variables. A computational scheme is proposed to build an orthogonal set of static modes so that the system matrices can have a diagonal or nearly diagonal form, providing thus high computational efficiency for application in the context of structural dynamics as well as flexible multibody dynamics. Connection to adjacent components is expressed through kinematic relationships between intrinsic variables and local displacements. The efficiency of the method is demonstrated on a simple example involving multiple unilateral contact. Copyright © 2015 John Wiley & Sons, Ltd.     

多体系统动力学研究进展

体系统动力学是当今力学领域的研究热点和难点之一,为解决机械、航空、航天、兵器、机器人领域中大量机械系     

An alternative complex variable formulation for an inclined crack in an orthotropic medium

Of concern in this paper is the study of the elastostatic fracture response of an orthotropic plate with an inclined crack and subjected at infinity to a biaxial uniform load. To this end an unconventional approach to the derivation of the complex variable expressions of the elastic fields is proposed. The above formulation has been used to solve the boundary value problem as superposition of Mode-I and Mode-II crack problems and it is shown that the near tip asymptotic expressions of stress and displacement fields are affected by non-singular terms originated by load biaxiality. The maximum tensile stress criterion is applied in order to investigate the effects of non-singular terms on the angle of crack extension.     

An effective simultaneous approach with variable time nodes for dynamic optimization problems

An effective simultaneous approach with variable time nodes is proposed to solve the dynamic optimization problems with multiple control components, where the variable time nodes for each control component are considered as parameters directly and the interval between the neighbouring variable nodes is further refined uniformly to ensure accuracy. Consequently, the method does not treat all the nodes as parameters to ensure efficiency. The gradient formulae and the sensitivities of the states with respect to the controls and the variable time nodes are further derived to solve the nonlinear programming problem transformed from the original dynamic optimization problem. The complete framework and detailed steps of the proposed method are also given. Two classic constrained dynamic optimization problems have been tested as an illustration, and detailed comparisons of the reported literature methods are carried out. The research results show the characteristics and the effectiveness of the proposed approach.     

An SPH shell formulation for plasticity and fracture analysis in explicit dynamics

This paper introduces a new modeling method suitable for the simulation of shell fracture under impact. This method relies on an entirely meshless approach based on the smoothed particle hydrodynamics (SPH) method. The paper also presents the SPH shell formulation being used as well as the different test cases used for its validation. A plasticity model of the global type throughout the thickness is also proposed and validated. Finally, in order to illustrate the capabilities of the method, fracture simulations using a simplified fracture criterion are presented. Copyright © 2008 John Wiley & Sons, Ltd.     

多体系统发射动力学及其应用

射击精度差、试验用弹量大、发射不安全是制约现代火箭、火炮武器发展的三大技术难题。武器系统精度和发射安全性取决于武器系统动力学规律,发射动力学作为研究武器系统发射过程中受力和运动规律的一门新兴综合工程学科,在国际上已成为提高火箭和火炮武器系统射击精度和发射安全性的新技术突破口,为射击精度和发射安全性设计与试验提供新理论与技术。文章研究多体系统发射动力学理论与技术及其在火箭和火炮武器射击精度和安全性设计与试验中的应用。     

An effective formulation for optimal scheduling of multistage multi-product batch plant based on due dates

A mathematical formulation for optimal scheduling of multistage multi-product batch plant with parallel units is presented. The allocations of tasks, units and stages are described by a set of binary variables. The sequence-dependent setup times of orders and the ready times of units are considered. Based on the due dates and the processing times of orders, the predefined processing sequence of orders is achieved. The continuous time representation mode is also used in the proposed model. The computations on several classic scheduling examples and comparisons with other existing models in the literatures demonstrate the proposed scheduling model can obtain the optimal solutions in a shorter time and has much fewer binary variables.     

Efficient sampling for spatial uncertainty quantification in multibody system dynamics applications

We present two methods for efficiently sampling the response (trajectory space) of multibody systems operating under spatial uncertainty, when the latter is assumed to be representable with Gaussian processes. In this case, the dynamics (time evolution) of the multibody systems depends on spatially indexed uncertain parameters that span infinite‐dimensional spaces. This places a heavy computational burden on existing methodologies, an issue addressed herein with two new conditional sampling approaches. When a single instance of the uncertainty is needed in the entire domain, we use a fast Fourier transform technique. When the initial conditions are fixed and the path distribution of the dynamical system is relatively narrow, we use an incremental sampling approach that is fast and has a small memory footprint. Both methods produce the same distributions as the widely used Cholesky‐based approaches. We illustrate this convergence at a smaller computational effort and memory cost for a simple non‐linear vehicle model. Copyright © 2009 John Wiley & Sons, Ltd.     

Two implementations of IRK integrators for real‐time multibody dynamics

Recently, several authors have proposed the use of implicit Runge–Kutta (IRK) integrators for the dynamics of multibody systems. On the other hand, Newmark‐type or structural integrators have shown to be appropriate when real‐time performance is demanded in that field. Therefore, the following question arises: might the IRK integrators be suitable for real‐time purposes? And, provided the answer is positive: might they be preferable to the Newmark‐type family? This paper reports an investigation which has been conducted by the authors in order to get insight into the two questions formulated above. Since, based on previous experiences, it can be suspected that the performance of the integrators may be dependent on the type of dynamic formulation applied, the following three formulations have been considered for the study: a global penalty formulation in dependent natural co‐ordinates (many constraints), a topological semi‐recursive penalty formulation in dependent relative co‐ordinates (few constraints), and a topological semi‐recursive formulation in independent relative co‐ordinates (no constraints). As representative of the IRK family, a two‐stage SDIRK integrator has been selected due to its low associated computational burden, while, on the side of the structural integrators, the trapezoidal rule has been chosen. Two alternative implementations have been proposed to combine the dynamic formulations and the SDIRK integrator. A very demanding maneuver of the whole model of a vehicle has been simulated through all the possible combinations dynamic‐formulation/integrator, for different time‐steps. Conclusions have been drawn based on the obtained results, which provide some practical criteria for those interested in achieving real‐time performance for large and complex multibody systems. Copyright © 2005 John Wiley & Sons, Ltd.