Initial condition correction in multibody dynamics

A simple procedure is presented to correct initial conditions for the coordinates and velocities prior to performing a kinematic or forward dynamic analysis of multibody systems. Such corrections are crucial since slight amount of constraint violations at the start of any numerical integration of equations of motion can lead to erroneous results. The correction process is based on the well-known method of minimizing the sum-of-squares of adjustments in the coordinates or velocities. The process provides a solution that is closest to the estimated values. It should be a simple task to implement this methodology as a preprocessing step for any kinematic or forward dynamic analysis program regardless of the formulation. Commemorative Contribution.     

Absolute nodal coordinate plane beam formulation for multibody systems dynamics

A new plane beam dynamic formulation for constrained multibody system dynamics is developed. Flexible multibody system dynamics includes rigid body dynamics and superimposed vibratory motions. The complexity of mechanical system dynamics originates from rotational kinematics, but the natural coordinate formulation does not use rotational coordinates, so that simple dynamic formulation is possible. These methods use only translational coordinates and simple algebraic constraints. A new formulation for plane flexible multibody systems are developed utilizing the curvature of a beam and point masses. Using absolute nodal coordinates, a constant mass matrix is obtained and the elastic force becomes a nonlinear function of the nodal coordinates. In this formulation, no infinitesimal or finite rotation assumptions are used and no assumption on the magnitude of the element rotations is made. The distributed body mass and applied forces are lumped to the point masses. Closed loop mechanical systems consisting of elastic beams can be modeled without constraints since the loop closure constraints can be substituted as beam longitudinal elasticity. A curved beam is modeled automatically. Several numerical examples are presented to show the effectiveness of this method.     

Flexible multibody simulation using hybrid integration scheme

One issue in the dynamic simulation of flexible multibody system is poor computation efficiency, which is due to high frequency components in the solution associated with a deformable body. Standard explicit numerical methods should take very small time steps in order to satisfy the absolute stability condition for the high frequency components and, in turn, the computational efficiency deteriorates. In this study, a hybrid integration scheme is applied to solve the equations of motion of a flexible multibody system for achieving better computational efficiency. The computation times and simulation results are compared between the hybrid scheme and conventional methods. The results demonstrate that the efficiency of a flexible multibody simulation can be improved by using the hybrid scheme.     

The development of a sliding joint for very flexible multibody dynamics using absolute nodal coordinate formulation

In this paper, a formulation for a spatial sliding joint is derived using absolute nodal coordinates and non-generalized coordinate and it allows a general multibody move along a very flexible cable. The large deformable motion of a spatial cable is presented using absolute nodal coordinate formulation, which is based on the finite element procedures and the general continuum mechanics theory to represent the elastic forces. And the nongeneralized coordinate, which is related to neither the inertia forces nor the external forces, is used to describe an arbitrary position along the centerline of a very flexible cable. Hereby, the non-generalized coordinate represents the arc-length parameter. The constraint equations for the sliding joint are expressed in terms of generalized coordinate and nongeneralized coordinate. In the constraint equations for the sliding joint, one constraint equation can be systematically eliminated. There are two independent Lagrange multipliers in the final system equations of motion associated with the sliding joint. The development of this sliding joint is important to analyze many mechanical systems such as pulley systems and pantograph-catenary systems for high speed-trains.     

Cable installation simulation by using a multibody dynamic model

A major concern when installing the cables into the underground conduit is minimizing the tensile forces exerted on the cables as they are pulled. This knowledge makes it possible to avoid over conservative design practices and to achieve substantial saving during construction. A general computing algorithm of predicting the tensile force of the cable pulled through the underground conduit with an arbitrary configuration is presented in this paper, which is based on multibody system dynamic formulation. The presented multibody dynamic model for this problem consists of the cable, the underground conduit, and the interaction between the cable and the conduit. In this paper, the cable is modeled by the finite cable element based on an absolute nodal coordinate formulation. The interaction between the cable and the underground conduit is described by the Hertz contact theory. Numerical examples are presented to illustrate the effectiveness and efficiency of the proposed method for estimating the cable tension.     

A two-step approach for model reduction in flexible multibody dynamics

In this work, a two-step approach for model reduction in flexible multibody dynamics is proposed. This technique is a combination of the Krylov-subspace method and a Gramian matrix based reduction approach that is particularly suited if a small reduced-order model of a system charged with many force-inputs has to be generated. The proposed methodology can be implemented efficiently using sparse matrix techniques and is therefore applicable to large-scale systems too. By a numerical example, it is demonstrated that the suggested two-step approach has very good approximation capabilities in the time as well as in the frequency domain and can help to reduce the computation time of a numerical simulation significantly.     

A method for improving dynamic solutions in flexible multibody dynamics

This paper presents a method for improving dynamic solutions that are obtained from the dynamic simulation of flexible multibody systems. The mode-acceleration concept in linear structural dynamics is utilized in the proposed method for improving accuracy in the postprocessing stage. A theoretical explanation is made on why the proposed method improves the dynamic solutions in the context of the mode-acceleration method. A mode-acceleration equation for each flexible body is defined and the load term in the right hand side of the equation is represented as a combination of space-dependent and time-dependent terms so that efficient computation of dynamic solutions can be achieved. The load term is obtained from dynamic simulation of a flexible multibody system and a finite element method is used to compute dynamic solutions by quasi-static analyses. Numerical examples show the effectiveness of the proposed method.     

An improved dynamic modeling of a multibody system with spherical joints

A dynamic modeling of multibody systems having spherical joints is reported in this work. In general, three intersecting orthogonal revolute joints are substituted for a spherical joint with vanishing lengths of intermediate links between the revolute joints. This procedure increases sizes of associated matrices in the equations of motion, thus increasing computational burden of an algorithm used for dynamic simulation and control. In the proposed methodology, Euler parameters, which are typically used for representation of a rigid-body orientation in three-dimensional Cartesian space, are employed to represent the orientation of a spherical joint that connects a link to its previous one providing three-degree-of-freedom motion capability. For the dynamic modeling, the concept of the Decoupled Natural Orthogonal Complement (DeNOC) matrices is utilized. It is shown in this work that the representation of spherical joints motion using Euler parameters avoids the unnecessary introduction of the intermediate links, thereby no increase in the sizes of the associated matrices with the dynamic equations of motion. To confirm the efficiency of the proposed representation, it is illustrated with the dynamic modeling of a spatial four-bar Revolute-Spherical–Spherical-Revolute (RSSR) mechanism, where the CPU time of the dynamic modeling based on proposed methodology is compared with that based on the revolute joints substitution. Finally, it is explained how a complex suspension and steering linkage can be modeled using the proposed concept of Euler parameters to represent a spherical joint.     

First order sensitivity analysis of flexible multibody systems using absolute nodal coordinate formulation

Design sensitivity analysis of flexible multibody systems is important in optimizing the performance of mechanical systems. The choice of coordinates to describe the motion of multibody systems has a great influence on the efficiency and accuracy of both the dynamic and sensitivity analysis. In the flexible multibody system dynamics, both the floating frame of reference formulation (FFRF) and absolute nodal coordinate formulation (ANCF) are frequently utilized to describe flexibility, however, only the former has been used in design sensitivity analysis. In this article, ANCF, which has been recently developed and focuses on modeling of beams and plates in large deformation problems, is extended into design sensitivity analysis of flexible multibody systems. The Motion equations of a constrained flexible multibody system are expressed as a set of index-3 differential algebraic equations (DAEs), in which the element elastic forces are defined using nonlinear strain-displacement relations. Both the direct differentiation method and adjoint variable method are performed to do sensitivity analysis and the related dynamic and sensitivity equations are integrated with HHT-I3 algorithm. In this paper, a new method to deduce system sensitivity equations is proposed. With this approach, the system sensitivity equations are constructed by assembling the element sensitivity equations with the help of invariant matrices, which results in the advantage that the complex symbolic differentiation of the dynamic equations is avoided when the flexible multibody system model is changed. Besides that, the dynamic and sensitivity equations formed with the proposed method can be efficiently integrated using HHT-I3 method, which makes the efficiency of the direct differentiation method comparable to that of the adjoint variable method when the number of design variables is not extremely large. All these improvements greatly enhance the application value of the direct differentiation method in the engineering optimization of the ANCF-based flexible multibody systems.     

基于三维坐标修正的改进型3DDV-Hop定位算法

针对传统3DDV-Hop算法对未知节点定位误差较大的问题,研究人员提出了各类基于跳数、跳距计算方法进行改进的定位算法,但这些算法中对跳数、跳距计算的改进方法有待优化且未二次求精未知节点坐标;为此,提出一种基于三维坐标修正的改进型3DDV-Hop定位算法,该算法通过为节点设定3类通信半径以及跳距加权以减少跳数、跳距计算误差,并构建正方体交叉区域对未知节点坐标进行二次求精。通过实验结果对比分析可知：本文所提基于三维坐标修正的改进型3DDV-Hop定位算法对未知节点平均定位误差确有显著降低。     

基于多体力学的车辆动力学控制系统仿真及优化

利用多体分析软件ADAMS建立了多自由度汽车整车多体动力学仿真模型,并进一步简化为15自由度非线性模型,结合2自由度线性模型建立PID控制策略,进行了冰面单周正弦工况下的汽车操纵稳定性仿真试验研究,采用自适应模拟退火算法与非线性序列二次规划法相结合的组合优化方法对控制系统的控制参数进行了分析和优化.结果表明,该控制方法能够大幅度提高车辆的操纵稳定性和安全性,能够适应复杂的路面和行驶工况,取得了良好的效果.     

Validation of flexible multibody dynamics beam formulations using benchmark problems

As the need to model flexibility arose in multibody dynamics, the floating frame of reference formulation was developed, but this approach can yield inaccurate results when elastic displacements becomes large. While the use of three-dimensional finite element formulations overcomes this problem, the associated computational cost is overwhelming. Consequently, beam models, which are one-dimensional approximations of three-dimensional elasticity, have become the workhorse of many flexible multibody dynamics codes. Numerous beam formulations have been proposed, such as the geometrically exact beam formulation or the absolute nodal coordinate formulation, to name just two. New solution strategies have been investigated as well, including the intrinsic beam formulation or the DAE approach. This paper provides a systematic comparison of these various approaches, which will be assessed by comparing their predictions for four benchmark problems. The first problem is the Princeton beam experiment, a study of the static large displacement and rotation behavior of a simple cantilevered beam under a gravity tip load. The second problem, the four-bar mechanism, focuses on a flexible mechanism involving beams and revolute joints. The third problem investigates the behavior of a beam bent in its plane of greatest flexural rigidity, resulting in lateral buckling when a critical value of the transverse load is reached. The last problem investigates the dynamic stability of a rotating shaft. The predictions of eight independent codes are compared for these four benchmark problems and are found to be in close agreement with each other and with experimental measurements, when available.     

Forward dynamical analysis of flexible multibody systems using system-level model reduction

Fast simulation (e.g., real-time) of flexible multibody systems is typically restricted by the presence of both differential and algebraic equations in the model equations, and the number of degrees of freedom required to accurately model flexibility. Model reduction techniques can alleviate the problem, although the classically used body-level model reduction and general-purpose system-level techniques do not eliminate the algebraic equations and do not necessarily result in optimal dimension reduction. In this research, Global Modal Parametrization, a model reduction technique for flexible multibody systems is further developed to speed up simulation of flexible multibody systems. The reduction of the model is achieved by projection on a curvilinear subspace instead of the classically used fixed vector space, requiring significantly less degrees of freedom to represent the system dynamics with the same level of accuracy. The numerical experiment in this paper illustrates previously unexposed sources of approximation error: (1) the rigid body motion is computed in a forward dynamical analysis resulting in a small divergence of the rigid body motion, and (2) the errors resulting from the transformation from the modal degrees of freedom of the reduced model back to the original degrees of freedom. The effect of the configuration space discretization coarseness on the different approximation error sources is investigated. The trade-offs to be defined by the user to control these approximation errors are explained.     

Real-time inverse dynamics control of parallel manipulators using general-purpose multibody software

This work deals with the problem of computing the inverse dynamics of complex constrained mechanical systems for real-time control applications. The main goal is the control of robotic systems using model-based schemes in which the inverse model itself is obtained using a general purpose multibody software, exploiting the redundant coordinate formalism. The resulting control scheme is essentially equivalent to a classical computed torque control, commonly used in robotics applications. This work proposes to use modern general-purpose multibody software to compute the inverse dynamics of complex rigid mechanisms in an efficient way, so that it suits the requirements of realistic real-time applications as well. This task can be very difficult, since it involves a higher number of equations than the relative coordinates approach. The latter is believed to be less general, and may suffer from topology limitations. The use of specialized linear algebra solvers makes this kind of control algorithms usable in real-time for mechanism models of realistic complexity. Numerical results from the simulation of practical applications are presented, consisting in a “delta” robot and a bio-mimetic 11 degrees of freedom manipulator controlled using the same software and the same algorithm.     

Interface reduction of flexible bodies for efficient modeling of body flexibility in multibody dynamics

The floating frame of reference techniques is an established technique to incorporate flexibility in multibody models. The model dimension of the body flexibility models can be reduced by model reduction techniques such as Component Mode Synthesis (CMS) or Krylov subspace-based techniques, but the efficiency of these techniques is limited by the number of interface nodes in which the flexible body is or can be loaded. A common solution to this problem is condensing the different nodes of a given interface surface into a single node, which represents the net motion of the interface surface. Commercial finite element packages offer two modeling techniques to condense interface surfaces: rigid multipoint constraints and interpolation multipoint constraints. Rigid multipoint constraints will typically result in stiffness overestimation, whereas interpolation multipoint constraints will lead to an underestimation. Which approximation of both is most suitable depends on the application. However, the default definition of interpolation multipoint constraints does not allow generation of reduced body flexibility models for multibody models. This paper gives a theoretical background of the problem cause, and offers a practical solution. The two modeling techniques result in significantly different approximations of the body flexibility dynamics, as is shown in a numerical example.     

Analytical approach for the evaluation of the torques using inverse multibody dynamics

A mathematical model for the analysis of human motion is presented in this paper. This model is based on linkage dynamics in order to understand trajectory and internal moment of force coordination. Mobility at the base of the supporting limb is a critical factor in the freedom to fall forward. The approach used to state a coupled system of differential equations of motion consists in introducing the displacement of the center of mass together with the displacement of each segment of the body and to evaluate the final system as a whole. The resultant methodology is task independent. The main goal of this study is to assist the work of health care professionals in the determination of the torques at the joints generated to maintain the movement. The evaluation of a weighted average of all the forces has served as a basis for other authors to obtain the center of pressure. In this work, the resultant ground reaction force passes through the center of mass of the body system enabling the calculation of the location of this force.     

Shock response analysis of hard disk drive using flexible multibody dynamics formulation

Recent development of the shock analysis on the HDD is briefly reviewed. A flexible multi-body dynamics formulation is developed to simulate the shock response of the HDD. If one component in the HDD is changed, only mode shapes and frequencies of that component should be re-calculated and then used to obtain the system’s response. Steady state Reynolds equation is solved to obtain the air pressure on the slider and disk for various slider positions. An air pressure table is formed and used to model the non-linear air bearing during the simulation. Responses of flying height for different direction and shock duration time are analyzed. Results show that the flying state of the slider is more sensitive to the shock with shorter duration time.     

火炮振动分析与多体系统模型研究

研究火炮振动对于提高射击精度有重要意义. 冲击激励的特殊性和机械系统的复杂性,使火炮振动成为长期关注的经典问题,建立逼近火炮真实运动状态的动力学模型、分析掌握各种冲击作用下的振动特性,是火炮振动的核心研究分支. 本文在发展多刚体与多柔体系统动力学理论的基础上,研究了火炮动力学的 Lagrange 方程及 Guass 变分建模方法,并由其建立了火炮振动的刚性、刚柔耦合多体系统模型; 针对案例分别研究了火炮系统的冲击响应规律及固有振动特性,获得了火炮振动与其结构参数的内在联系. 研究结果对于改善火炮的振动性能、提高射击精度,具有理论指导意义和工程应用价值.