## Representation of multibody systems

The definition of multibody systems is a complex task: all the properties of the system must be defined to provide a complete, consistent representation of the problem. Among the basic properties that must be defined are 1) the connectivity of the system, *i.e.* how the various components of the system are connected together, 2) the geometry of the problem, *i.e.* the dimensions and shape od the various components and 3) the structural elements of the system and their physical properties, such as their mass and stiffness characteristics. These different aspects of the problem are clearly distinct, and hence, it convenient to use different representations to define each aspect of the problem.

##### Figure 1. Relationship between topological entities,

geometric components, and structural elements.

The first step in the definition of a complex mechanical system is the definition of its connectivity, which is considerably eased by making systematic use of basic principles of topology. The first column of fig. 1 shows the basic topological entities: *vertices*, *edges*, *faces*, and *blocks*. Note that it is important to make a clear distinction between topology and geometry: the system topology defines the connections among the various elements of the system, whereas the system geometry deals with the dimensions and shapes of its components. Topological entities are hierarchical in nature. The edge points to two distinct vertices. The face points to four edges, each of which points to two vertices; of course, two adjacent edges share a common vertex and hence, a face involves four distinct vertices only. Finally, a block points to six faces, each of which points to four edges, themselves each pointing to two vertices. Clearly, two adjacent faces share a common edge, and three intersecting edges share a common vertex; hence, the block involves twelve distinct edges and eight distinct vertices only.

While topology and geometry are clearly distinct concepts, they are also closely related to one another; the dictionary definition of the word topology is “The study of the properties of geometric figures or solids that are not changed by homeomorphisms, such as stretching or bending. Donuts and picture frames are topologically equivalent, for example.” The second column of fig. 1 shows the basic geometric components: *points*, *curves*, *surfaces*, and *volumes*, and their relationship with the corresponding topological entities. A point is defined by its three Cartesian coordinates in a given reference frame. Curves and surfaces are defined by their NURBS representation, which involves a number of control points, also defined by their Cartesian coordinates. Additional geometric components are often required for the definition of multibody systems. The most important is probably the *triad*, *i.e.* a set of three mutually orthogonal unit vectors, which is denoted *E* = (e_{1}, e_{2}, e_{3}), where e_{1}, e_{2} and e_{3} are the three mutually orthogonal unit vectors. Another important geometric component is the *fixed frame* of reference, which consist of an origin point and a triad, and is denoted *F* = (**A**, *E*), where **A** is the origin point of the fixed frame and *E* the triad defining its orientation.

The relationship between topology and geometry has been underlined in the previous paragraph. The third column of fig. 1 shows how basic structural elements are related to topological entities and geometric components. Vertices will be associated with point masses, Edges will be associated with one-dimensional (1D) structural elements such as beams or cables, faces with two-dimensional (2D) elements such as plates and shells and blocks with three-dimensional (3D) elements such as 3D elasticity elements. Boundary conditions can be applied to vertices, edges, or faces. Structural elements are associated with a topological entity, and inherit their geometric features from the geometric component associated with the topological entity. For instance, if a beam is associated with an edge featuring an associated curve, this curve will then define the spatial geometry of the beam. However, other types of structural elements could also be associated with edges; consider, for instance, a revolute joint that allows the relative rotation of two structural elements of the model. In this case, the revolute joint is associated with an edge featuring an associated triad, and this triad then defines the axis about which the relative rotation is allowed.

In view of the above discussion, the definition of a multibody system proceeds with the following steps.

- Define the topological entities of the system. The entities to be defined are vertices, edges, and faces. Each topological entity is associated with one or more geometric components.
- Define the geometric objects of the system. The components to be defined are points, curves, and surfaces. To facilitate the definition process, any of these geometric components ca be defined in a local fixed frame of reference.
- Define the structural elements and constraint elements of the system. Structural elements include beams, cables, or shells, among others. Constraint elements include the many types of holonomic constraint commonly found in multibody system, such as revolute joints, universal joints, or prismatic joints, among others.

Once these three elements of the model have been defined, the basic configuration of the multibody system is in place. Of course, additional information will be required, such as the physical properties of the structural elements, analysis control parameters, or postprocessing requirements, among many others. However, the three step process outlined above provides a rational approach to the definition of complex mechanical systems. It clearly delineates three distinct aspect of the system: topological entities, geometric components, and structural elements. When the database used to represent the complex mechanical systems is build on these entities, coding is considerably simplified, and furthermore, the topological configuration of the mechanical system provides a rational foundation for the development of a graphical user interface for data input.

## Example

To illustrate the concepts discussed in the previous section, a specific example will be presented here. Consider the four bar mechanism problem depicted in fig. 2. ** Beam1** is of length L

_{1}= 0.12 m and is connected to the ground at point

**A**by means of a revolute joint, denoted

**.**

*RvjA***is of length L**

*Beam2*_{2}= 0.24 m and is connected to

**at point**

*Beam1***B**with a revolute joint, denoted

**. Finally,**

*RvjB***is of length L**

*Beam3*_{3}= 0.12 m and is connected to

**and the ground at points**

*Beam2***C**and

**D**, respectively, by means of two revolute joints, denoted

**and**

*RvjC***, respectively. In the reference configuration, the bars of this planar mechanism intersect each other at 90 degree angles and the axes of rotation of the revolute joints at points**

*RvjD***A**,

**B**, and

**D**are normal to the plane of the mechanism. However, the axis of rotation of the revolute joint at point

**C**is at a 5 degree angle with respect to this normal to simulate an initial defect in the mechanism. A torque is applied on

**at point**

*Beam1***A**to enforce a constant angular velocity Ω = 20 rad/sec for

**. If the beams were infinitely rigid, no motion would be possible as the mechanism locks. For elastic beams, motion becomes possible, but generates large internal forces.**

*Beam1***has the following physical characteristics: axial stiffness EA = 40 MN, bending stiffnesses H**

*Beam1*_{22}= H

_{33}= 0.24 MN.m

^{2}, torsional stiffness GJ = 0.28 MN.m

^{2}, and mass per unit span m = 3.2 kg/m.

**and**

*Beams2***have the following physical characteristics: axial stiffness EA = 40 MN, bending stiffnesses H**

*3*_{22}= H

_{33}= 24 kN.m

^{2}, torsional stiffness GJ = 28 kN.m

^{2}, and mass per unit span m = 1.6 kg/m.

##### Figure 2. The four bar mechanism problem.

The above description of the system clearly involves the three types of information required to define the system. The connectivity of the various components is given: the beams are interconnected by revolute joints at their end points. The geometry of the system of also given: the length and orientation of the beams are defined, as well as the orientations of the axes of rotation of each revolute joint. The structural elements of the system are defined: in this example, the system solely consists of beams and revolute joints. Of course, the physical characteristics of the various structural elements are also defined.

##### Figure 3. Topology of the four bar mechanism problem.

Figure 3 now depicts the topology of the four bar mechanism. ** Beam1**,

**and**

*Beam2***are now associated with three edges, denoted**

*Beam3***,**

*EdgeBeam1***and**

*EdgeBeam2***, respectively. Similarly, the four revolute joints of the system,**

*EdgeBeam3***,**

*RvjA***,**

*RvjB***and**

*RvjC***, becomes four edges, denoted**

*RvjD***,**

*EdgeRvjA***,**

*EdgeRvjB***and**

*EdgeRvjC***, respectively. For practical reasons, it is important to define the relative rotations at the intersection of the bars at points**

*EdgeRvjD***A**,

**B**,

**C**and

**D**, and edges

**,**

*EdgeRotationA***,**

*EdgeRotationB***and**

*EdgeRotationC***, respectively, are used to defined these relative rotations. Finally, edge**

*EdgeRotationD***is used to prescribe the relative rotation at point**

*EdgePrdA***A**to the desired 20 rad/sec value.

The various edges are each connected to two vertices, and the connectivity of the system then naturally follows from the fact that two or more edges are connected to identical vertices. For instance, ** EdgeRvjB** shares a vertex,

**, with**

*VertexB2***, and another vertex,**

*EdgeBeam2***, with**

*VertexB1***. This implies that revolute joint**

*EdgeBeam1***interconnects**

*RvjB***and**

*Beam1***. Note that vertices**

*Beam2***,**

*VertexB1***and**

*VertexB2***are at the same geometric point,**

*VertexB3**i.e.*the geometric points associated with these vertices share the same Cartesian coordinates. Boundary conditions are applied at

**and**

*VertexA1***.**

*VertexD2*It is interesting to contrast the topological representation of the system given in fig. 3 with its geometric representation shown in fig. 2. While the geometric representation of the system obviously provides insight into the geometric elements of the system such as the shape and dimensions of the various components, it also hides important features of the system such as the connection between its elements. Clearly, both topological and geometric representations of the system are equally useful to understand the configuration of the system.