Multibody Systems

Thus, the horizon of supramolecular chemistry lies on the road towards complexity, from the single molecule towards collective properties of adaptive multibody systems of interacting components. 

W. Schiehlen (ed.) Advanced Multibody System Dynamics. Simulation and Software Tools. 

J.-C. Samin and P. Fisette Symbolic Modeling of Multibody Systems. 2003... 

Fig. 3.8 Dielectric elastomer ocean wave power generator based on a an articulated multibody system, individual roll transducer (top left), concatenated rolls in a generator module (bottom left), and buoy at sea trial site (right)...

Physical motion is common to most situations in which the human functions and is therefore fundamental to the analysis of performance. Parameters such as segment position, orientation, velocity, and acceleration are derived using kinematic or dynamic analysis or both. This approach is equally appropriate for operations on a single joint system or linked multibody systems, such as is typically required for human analysis. Depending on the desired output, foreword (direct) or inverse analysis may be employed to obtain the parameters of interest. For example, inverse dynamic analysis can provide joint torque, given motion and force data while foreword (direct) dynamic analysis uses joint torque to derive motion. Especially for three-dimensional analyses of multijoint systems, the methods are quite complex and are presently a focal point for computer implementation [Allard et al., 1994]. 

Linked multibody system A system of three or more individual segments joined (as an open or closed chain) in some manner with the degree of freedom between any two segments defined by the characteristics of the corresponding hinge. 

Multibody Formulations Both abovementioned formulizations can be described by the following common form that is usually used nowadays to describe the so-called multibody systems ... 

As an example, consider the motion of an arm-forearm system illustrated in Fig. 7.1. The corresponding equation of motion for the elbow joint (point C), or a two-link, multibody system is given in Eq. (7.1). 

The energy method approach uses Lagrange s equation (and/or Hamilton s principle, if appropriate) and differs from die newtonian approach by the dependence upon scalar quantities and velocities. This approach is particularly useful if the dynamic system has several degrees of freedom and the forces experienced by the system are derived from potential functions. In summary, the energy method approach often simplifies the derivation of the equations of motion for complex multibody systems involving several degrees of freedom as seen in human biodynamics. 

For a multibody system, each body would require a kinematics table and a corresponding schematic. The following examples illustrate the steps required for solving problems by the table method. Note that one example includes the expressions for acceleration to demonstrate the use of the table method with the Newton-Euler approach, while all other examples consider only the velocity. 

If the multibody system represented in Fig. 7.6 is considered to represent a human torso, upper arm, forearm, and hand, then Table 7.5 results in the velocity at the shoulder (point B) express in the body-fixed coordinate system of the torso segment, a 2, a,. Similarly, the b bz, bj coordinate system is body-fixed to the upper arm segment, the c, C2, C3 system to the forearm segment, and... 

Bos, A. M. (1986) Modelling multibody systems in terms of multibond graphs with application to a motorcycle. Ph.D. Thesis. The Netherlands University of Twente Enschede. 

The primary objective of this bode is the concise derivation and clear presentation of efficient algorithms for the dynamic simulation of robotic systems. In particular, robots and other multibody systems with closed-loop kinematic configurations are considered. It is assumed that each robot cr muldbody system is comprised of rigid bodies which are connected by ideal Joints and powered by ideal actuator. 

In an attempt to circumvent the computational complexity of matrix inversion, some researchers are pursuing solutions for the joint accelerations which have a linear recursive form. The inversion of the in tia matrix is explicitly avoided. The resulting linear recursive algorithms have a reduced computational complexity which is 0(N). This is the second basic solution approach to the Direct Dynamics problem, and it has been rqjplied to serial open chains [3,7], single closed chains [22], and some more genoal multibody systems [4, 37]. It is believed that the structure of linear recursive algorithms may also facilitate their implementation on parallel computer systems. 

To include general joints and contacts with multiple degrees of freedom in a multibody system, an extended model of the interconnections and interactions between individual bodies of that system is required. This section will summarize the important features of one such description, that of Roberson and Schwertassek [36], which is consistent with the invariant method discussed in [26]. This particular model is also used by Brandi, Johanni, and Otter in [3]. The notation used here is slightly different from that found in [3] in order to tiuuntain consistency in the presented algorithms. This model will be used extensively throughout this book in the development of all algorithms. 

In general, the inertia matrix of a manipulator defines the relationship between certain forces exerted on the system and some corresponding acceleration vector. This relationship is of great importance both in real-time control and in the simulation of multibody systems. In the control realm, for example, the inertia matrix has been used to decouple robot dynamics so that control schemes may be more effectively tq>plied [19]. This may be accomplished either in joint space O operational space, since the inertia matrix may be defined in eith domain. The inertia matrix has also been used in the analysis of collision effects [43]. In addition to its use in control applications, the inertia matrix is an explicit and integral part of certain Direct Dynamics algorithms which are used to solve the simulation problem for manipulators and other multibody systems [2, 8, 31, 33,42]. 

H. Brandi, R. Johanni, and M. Otter. An Algorithm for the Simulation of Multibody Systems with Kinematic Loops. In Proceedings of the IFToMM Seventh World Congress on the Theory of Machines and Mechanisms, Sevilla, Spain, September 1987. 

Lubich, C., Nowak, U., Pohle, U., Engstler, C., MEXX - Numerical Software for the Integration of Constrained Mechanical Multibody Systems, Preprint SC 92-12, Konrad-Zuse-Zentrum Berlin (1992)... 

Other engineering systems that are suitably modelled by means of bond graphs are multibody systems composed of bodies assumed rigid that are interconnected by different types of joints. Examples are industrial robots or mobile systems such as walking machines. Bond graph models of multibody systems are conveniently represented in the form of multibond graphs. Their advantage is that they enable a clear and concise presentation of multibody systems. 

Chapter 9 gives a review on how multibody systems can be modelled by means of multibond graphs. A major contribution of the chapter is a procedure that provides a minimum number of break variables in multibond graphs with ZCPs. For the state variables and these break variables (also called semi-state variables) a DAE system can be formulated that can be solved by means of the backward differentiation formula (BDF) method implemented in the widely used DASSL code. The approach is illustrated by means of a multibond graph with ZCPs of the planar physical pendulum example. 

The method of multibody systems allows the dynamical analysis of machines and structures, see e.g. (Schiehlen Eberhard 2004) and (Schiehlen, Guse 8c Seifried 2006). More recently contact and impact problems featuring unilateral constraints were considered too, see (Pfeiffer Glocker 1996). A multibody system is represented by its equations of motion as... 

Schiehlen, W. Seifried, R. 2004. Three approaches for elastodynamic contact in multibody systems. Multibody System Dynamics, 12 1-16. 

Seifried, R., Hu, B. Eherhard, P. 2003. Numerical and experimental investigation of radial impacts on a half-circular plate. Multibody Systems Dynamics, 9 265-281. 

Schiehlen and Seifried (Chapter 9) elaborately describe the impact on beams that resnlts in large rigid body motions and small structural waves. Such mechanical systems are often modeled as multibody systems to describe the large nonlinear motion where the impacts are treated by the coefficient of restitution. The coefficient of restitution is considered as deterministic number depending on the material, the shape and the... 

After a short introduction to multibody systems and the mathematical formulation of the equations of motion, different numerical methods used to solve simulation tasks are presented. The presentation is supported by a simple model of a truck. This truck model will follow the reader from the title page to the appendix in various versions, specially adapted to the topics. 

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