And everything is connected to everything

In a multi-body system, individual bodies are connected to each other with joints. Examples are complex structures such as industrial robots and crank mechanism in combustion engines, to name just two characteristic applications.

In the following, the dynamics of rigid bodies will be mentioned first. After the connection of single bodies to complex systems is described, the extension to flexible multi-body systems is following. The reason for this is that multi-body system simulation was initially developed for rigid-body systems. In addition, even today the individual components of numerous systems can be modelled in good approximation as rigid bodies.



An unsupported rigid body has six degrees of freedom in the spatial case, three degrees of displacement and three degrees of rotational freedom. They are also called generalized coordinates. According to Newton's second axiom, the movements of a rigid body are defined by the directions of the acting forces and moments and influenced by mass inertia. The inertias act against the directions of movement and describe the ability of a body to want to remain in a state of rest or uniform movement.



The movements of the individual bodies of a multi-body system are related with kinematic equations. The special forms of the kinematic equations depend on the location and type of joints that connect the individual bodies. In addition to ideal joints, springs and dampers are also used to model the flexibilities and friction effects in the connection points. The motion of complex structures can be numerically analyzed in a very short time with only six degrees of freedom per rigid body; this also enables real-time simulations, which are used in control technology.



Typical applications where multi-body systems are used successfully are chassis and transmissions of all types. A special application is the analysis of the swash plate mechanism in main rotor heads of helicopters. It controls the alignment of the rotor blades, whose flexibility can be easily modelled with finite elements. If the number of degrees of freedom is to be kept small, methods of model reduction can be used, which lead to dominant modes of vibration. They can be considered with their generalized coordinates in the system of equations of motion.


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