Multibody system

A multi-body system is a mechanical system of individual bodies, to each other by joints or force elements (eg springs, dampers ) are coupled and are under the influence of forces.

• 5.1 Quadratic velocity vector
• 5.2 Lagrangian multipliers
• 5.3 Constraints and joints
• 5.4 Minimal coordinates

Introduction

Using the kinematics, a section of the mechanics, the motion of the body is formulated mathematically. Although a multi-body system can also be immobile, is usually described as a form a system whose parts can move relative to each other. The investigation of movement ( dynamism ) of a multi-body system is called multi-body dynamics.

In this article an overview of some important aspects of multibody systems is given. However, in the late 1990s, many newer subdivisions developed ( optimization, sensitivity analysis, robotics, control, automotive and railway vehicles, and others), which did not find place in this article.

Areas of application

Multibody systems are used for modeling the movement of (sub - ) objects for technical applications

• Robotics
• Vehicle simulation ( dynamics, tires, improve comfort, weight reduction, and others)
• Simulation of motors, gears, chain drives, belt drives, including
• Simulation of Carrier, especially paper machines
• Particle simulation ( eg sand)
• Biomechanics ( movement optimization, prosthesis, crutches, and others)
• Aerospace industry (eg start / landing maneuver of aircraft, rotor of a helicopter )
• Simulation of ships and other floating bodies ( hydrodynamic coupling through waves)
• Space research (eg, satellite )
• Military applications

Example

The following figure shows a typical example of a multi-body system. This system is also called crank mechanism ( slider - crank English ). In this example, a rigid body is used for the driven crank which drives a flexible rod, and finally moves the end mass ( mass of concentrated ). The final mass can only move within the guide. There are installed three rotary joints, one between the crank and inertial frame, one between the crank and connecting rod and connecting rod between one and final mass.

Terms

Under a body you (not to be confused with the mathematical body or human body ) means a rigid or flexible body in the mechanical sense. A body is here, for example, an arm of a robot, the steering wheel of a car, but also the forearm of a human. Joints make it mobile connections between bodies. One can imagine in a machine or even in a car, for example, on the basis of a joint in the human body or the example of a joint.

In the multi- body dynamics are two other concepts of central importance: degree of freedom and constraint.

Degree of freedom

In terms of a mechanical body represents the number of degrees of freedom, the number of independent movement possibilities represents a rigid body has six degrees of freedom in general spatial motion, 3 translational degrees of freedom and three rotational degrees of freedom. If only the motion of a body (and all its points ) in a plane, so that the body has only three degrees of freedom: two translational degrees of freedom and one rotational degree of freedom.

When viewing a body in space, such as the computer mouse, the three translational degrees of freedom can be simply going through the motions left-right, front - back, up-down pose. The three degrees of freedom of rotation of the body (the computer mouse ) will be described by rotation about the axis along which the translation proceeds (eg, about the longitudinal axis of the mouse while the fore-aft translation ).

Constraint

Constraints represent a restriction of movement possibilities of bodies dar. constraints can be applied either between two bodies or between a body and a fixed point in space. Contrary to the assumption in the link " constraints " can not only mass points subject to a constraint. A constraint can be applied in a multi-body system and on rotations, velocities ( and angular velocities) and accelerations.

Equations of motion

The movement of the multi-body system is described by the equations of motion resulting from the second Newtonian axiom and the additional constraints.

The motion of a rigid body system can be expressed as follows:

This formulation is also designated by the term redundant coordinates. These represent generalized coordinates degrees of freedom not under compulsion body is, is the mass matrix, which may depend on the generalized coordinates, describes the constraints and the derivation of the constraints on the generalized coordinates. The symbol referred to in this formulation of the equations of motion of the Lagrange multipliers. Assuming a single body, so you can see the generalized coordinates divided into

The translations describes, and describes the rotations.

Square velocity vector

The expression represents the square speed vector which, according to results in the derivation of the equations of the kinetic energy. This term depends on the chosen rotation parameters.

Lagrangian multipliers

The Lagrangian multipliers are each associated with a constraint and provide mostly force or torque is acting in the direction of the locked degree of freedom, but no work.

Constraints and joints

Joints are expressed in the most general description language for multi-body systems using constraints. As already mentioned, constraints can be used both for displacement and rotation as well as the time derivatives of these quantities.

One moment differs holonomic and non- holonomic constraints, a more detailed description is given constraint in the section. For multibody systems, it is important that common joints such as cylindrical joint, ball joint, prismatic joint, etc. fall under holonomic constraints.

Minimal coordinates

The equations of motion are described with redundant coordinates, which are not independent of each other due to the constraints. It is possible under certain conditions, that this formulation in a system with non-redundant, ie independent coordinates and without constraints, overwrites. This transformation is generally not possible when the associated body having a closed ring (loop) or if it is not simple holonomic joints. Nevertheless, one can go on to a system with the smallest possible number of coordinates by inserting constraints only on certain unavoidable points and at the other points used non-redundant coordinates. One possible type of formulation with minimal coordinates is called the recursive formulation.

English technical terms

Usually, multi- body system is translated with multibody system or multi -body system into English, and multi-body dynamics and dynamics of multibody systems or multi- body system dynamics. Constraint is called the constraint condition, joints are denoted by joints. As is clear from the terminology, is a multi- chamber system of single bodies, which are referred to as "Bodies ". Each of said body comprises at least one " marker ", that is a point of attack for forces or joints. By means of these joints ( " joints" ) can be connected together the individual bodies, each body has exactly one " joint " because its position and type of connection in the entire system must be defined